HEART AND LUNG, SURGERY OF

since 191to- new surgical work on the heart has been mainly devoted to three subjects. i. surgery of the heart removal of foreign bodies retained in the heart wall.—the safety of the methods of surgical approach to the heart has been further demonstrated. surgeons in many countries have re- moved war missiles from the wall of the heart and from the peri- cardium with a surprisingly small mortality. r.le fort (bull. et mem. acad. de med., vol. 80, p. 147, 1918) removed 11 foreign bodies from nine patients, with one death. roberts successfully removed a bullet from the posterior wall of the jeft ventricle, which had been present for seven years and was causing cardiac disturbance. the approach by resection of one rib and traversing the pleura was excellent. he had also removed four fragments of shell from the pericardium in four patients by the same route, with no mortality. operations for stenosis of the valvular orifices of the heart.— a new ficld for surgery in the treatment of what has for long been a purely medical subject is being opened up by work which is at the present moment largely experimental. the subject is still in its infancy, but enough has been done to show that with further experience certain selected cases of mitral stenosis, in which the musculature of the heart has not been too much dam- aged, may be submitted to operation with a fair prospect of improvement. duff s. allen and e. a. graham (jour. amer. med. assoc., sept. 23 1922 and rch. of surg., jan. 1924) have 325 invented an instrument which they call a cardioscope. this is a tube, closed at one end by a lens, which is introduced into the heart through a small incision. when the lens is pressed against the heart wall so that no blood intervenes, the endocardium, illuminated by a small electric bulb, is clearly visible. in the wall of the tube is fitted a rod ending in a small knife, with which a stenosed valve may be incised under direct vision. in dogs, when the incision to introduce the cardioscope was made in the auricular appendix, all the dogs recovered. e. c. cutler and s. a. l. levine (the boston afed. and surg. jour., june 28 1923), believing that a mere incision in the stenosed ring will heal without relief of the stenosis, invented an instrument which punches out a piece of tissue. they have used this successfully in the case of a girl, aged 12, suffering from mitral stenosis. she was alive 1o months later and was still improving. h.s. souttar (brit. afed. jour., oct. 3 1925) reports a brilliant operation on a girl of 15 with mitral stenosis and regurgitation. he exposed the heart by a flap operation, and placed a light clamp on the base of the auricular appendix. aiter two guide sutures had been inserted, the appen- dix was incised and drawn over the finger like a glove, haemor- rhage being thus obviated. the interior of the auricle was easily explored by the finger, no effect on the pulse being noted. the blood pressure instantly fell to zero as the passage of blood through the orifice was abolished. souttar intended to divide the stenosed orifice with a knife passed along the finger, but as the stenosis was found to be of moderate degree with little thickening of the valve, he contented himself with stretching the orifice with the finger. the appendage was ligated at its base and the wound of the chest wall closed. the patient's con- dition was improved and she was well three months later, though still somewhat breathless on exertion. cervical sympathectomy for angina pectoris— the cause of this very distressing disease is not settled with certainty. but as changes in the cardiac muscle, coronary arterics and aorta are usually present it may be considered. jonnesco, of bucharest, in 1916 first performed the removal of the left cervical sympathetic chain, including the upper and middle cervical ganglia and the first thoracic ganglion. the result, followed for four years, was complete relief from symp- toms. ‘t. jonnesco operates under spinal anaesthesia (la presse med., april 26 1922). w. b. coffey and p. k. brown (arch. int. med., vol. 31, p. 200, 1923) report six cases with one death and great improvement in the other five, h. lillienthal carch. of surg., vol. 10, p. §31, 1925) three cases. in one, the © cervical sympathetic was removed on both sides. all were cured. h. h. kerr (ann. of surg., vol. 82, p. 354, 1925) reports five cases. he removes the superior cervical ganglion only, under local infiltration anaesthesia by novocain, and obtains the same results as the others by the more formidable operation per- formed by jonnesco. il. surgery of the lung since the world war the surgery of the lung has made great strides, and in all countries is engaging the attention of active workers. while differential pressure anaesthesia of some kind is still advisable, the introduction of intra-tracheal insufflation has enabled the elaborate and costly pressure chambers to be dispensed with. for,.many types of operation ordinary imhala- tion anaesthesia, especially by nitrous oxide and oxygen, suf- fices. local infiltration anaesthesia, combined with nerve block- ing by novocain or some similar drug, is largely used, and for some operations is essential. it is being combined with hght analgesia by nitrous oxide and oxygen, or even ether by some surgeons in the case of nervous patients, for whom deep general anaesthesia is inadvisable (see anaestitetics). surgical treatment of pulmonary tuberculosis —direct opera- tions on the tuberculous lung, cither by drainage of tuberculous cavities or the resection of one or more lobes of the lung, have been abandoned by the majority of thoracic surgeons. the great advance in latter years has been in devising methods of collapsing the affected lung either temporarily or permanently. 326 principles of collapse therapy.—in all other parts of the body it is found that rest of the affected part is beneficial, but in the case of the lung the respiratory movements, exaggerated by coughing, continue unless the lung is compressed. by collapse, stasis in the lymphatic vessels occurs, preventing spread of tubercle bacilli to other parts of the lung and diminishing the entrance of toxins into the general circulation with consequent improvement of the general resistance of the body. congestion of the collapsed lung with blood appears to occur, favouring the formation of fibrous tissue and healing of the lesions. me- chanically, the effect is to allow the retraction of the new formed fibrous tissue and to collapse cavities and dilated bronchi. this prevents the retention of secretions, which often become second- arily infected with other organisms, and allows the walls of the cavities to come in contact and so to heal. this retraction occurs normally to a limited extent by the pulling up of the diaphragm, the pulling over of the mediastinum, and the assumption of the expiratory position by the ribs, but falls far short of that neces- sary for complete obliteration of the cavities. methods of producing collapse. —artificial pneumothorax was first proposed by james carson, of liverpool, in 1821, but was introduced into practice by forlanini, of pavia, in 1882 (gazz. d. osp., 1882). it is now extensively practised by physicians all over the world. by the introduction of air, oxygen or nitrogen through a hollow needle into the pleural cavity the lung may be more completely collapsed.than by any other method. as this method is simple and satisfactory it is the method of choice, and other operations are only undertaken when this is impossible, through the presence of adhesions binding the two layers of the pleura together (see pneumothorax). division of adhesions —where the adhesions are few and bandlike, so that a partial collapse only can be obtained, it is necessary to divide them. the operation of thoracotomy and division of the bands by direct vision is unnecessarily severe. jacobaeus, of stockholm, in 1913 devised an instrument similar to a cystoscope, which he calls a thoracoscope. after the induc- tion of a pneumothorax this is introduced under novocain anaes- thesia through a canula. through a smaller canula a galvano- cautery is introduced. thus the adhesions may be burnt through under the vision of the operator without opening the chest cavity (ii. c. jacobacus, proc. roy. soc. med., 1922-3, vol. 16, p. 45). by using the cautery at a red heat only, haemorrhage is avoided. the adhesions often pull out a cone of lung substance in which may be a prolongation of a lung cavity, so that it is ’ important to burn the adhesions through as near the chest wall as possible. paralysis of the diaphragm.—stuertz first proposed this opera- tion by division of the phrenic nerve in the neck, which was independently devised by sauerbruch. the result of depriving the diaphragm on one side of its motor innervation is that it rises in the thorax as much as 23 to three in., thus allowing the lower lobe partially to collapse and putting it at rest. the operation is easily done in a few minutes under local anaesthesia by novocain, the nerve being found running obliquely across the scalenus anticus muscle in the neck. at first the nerve was simply divided, but it was found that accessory fibres joined it below the point of section in 20-30% of individuals. w. felix’s operation of exairesis ’ (deutsch. zisch. f. chir., no. 171, p. 283, 1922) has gained favour. in this modification the lower end of the divided nerve is seized in forceps and gradually drawn out of the thorax. sometimes the whole length of the nerve as far as the diaphragm is successfully extracted, but in any case the collaterals which join it in its upper part are also severed so that the paralysis is complete. the operation is used where basal adhesions are preventing full collapse in pneumothorax, as an addition to thoracoplasty, in some cases of early tuberculosis of the lower lobe only and for the relief of pain in the region of the diaphragm on coughing. extrapleural pneumelysis—the operation consists in separat- ing the parietal layer of the pleura from the chest wall over a limited area and filling the space thus formed with some ma- terial which is not absorbed, thereby compressing the lung heart and lung, surgery of beneath. a portion of one rib is resected and the posterior peri- osteum and endothoracic fascia are carefully incised. the gloved finger then separates the outer surface of the parietal pleura from the inner surface of the endothoracic fascia. as the air which enters the cavity thus formed is rapidly absorbed it is necessary to provide some filling. tuffer, in rg1o, first used fat obtained from omentum ora lipoma. this has become one of the favourite methods, the fat being obtained from the patient’s thigh (tuffer, bull. et mem. soc. de chir. de paris, vol. 49, 1249, 1923). ifa very careful aseptic technique is not followed the fat may be extruded. owing to the difficulty of obtaining enough fat, baer, in 1913, used a paraflin filling (it. baer, afiinch. med. wochen- schr., vol. 68, 1921). this is readily available but is heavy, and tends to be extruded later in many cases. e. w. archibald, of montreal (am. rev. of tuberc., vol. 4, p. 828, 1921), has used a pedicled muscle graft obtained by detaching the pectoral muscle. the operation is almost confined to apical lesions and is used either alone, or following thoracoplasty, in cases where the apex of the lung is incompletely collapsed. it may also be of service where the apex alone is adherent and a satisfactory pneumo- thorax has collapsed the lower part of the lung. extrapleural thoracoplasty—collapsing operations on the chest wall were suggested as long ago as 1888 (quincke) and 18g0 (spengler). in 1907 brauer and friedreich removed large portions of the ribs in their lateral portions. wilms, in rgrr, removed portions both posteriorly and anteriorly. the modern operation of paravertebral thoracoplasty is due to sauerbruch and the scandinavians: bull, of osol; saugman, of veilefiord; jacobaeus and key, of stockholm. the operation may be done in one, two or more stages and consists essentially in resecting a portion of the first to the roth or 11th ribs posteriorly through a long incision posterior to the scapula and turning forwards along the roth rib. the muscles attached to the scapula are divided and the scapula is turned forwards. it is essential that the mbs should be resected as far back as the tips of the transverse proc- esses of the vertebrae. the length of rib removed varies from 4 in. of the first rib to 7 or 8 in. of the middle and lower ones, the operation is a severe one, but the mortality, which was at first about 8°% has been reduced by proper selection of cases to about 2% in the hands of experienced operators. many opcra- tors still prefer a local anaesthesia by novocain, but the tendency now, except in cases with profuse sputum, is to operate under general anesthesia by nitrous oxide and oxygen preceded by an injection of morphine and hyoscine. ‘the results of these opera- -tions have shown that in cases of unilateral or mainly unilateral pulmonary tuberculosis which are not improving under sana- torium treatment a new field of hope is opening out. f. sauer- bruch (chir. der brustorgane, 1920), h. c. jacobaeus and e. key (acta chir. scand., 1923); p. bull (proc. roy. soc. afed., 1924, vol. 17, p. 1); j. alexander (surg. of pulmon. tub., 1925); j. gravesen (surg. treat. of pulmon. and pleur. tub., 1925). bronchiectasis.—three types of operation are used in the treatment of this distressing condition: (1) drainage operations, (2) operations to collapse the lungs, (3) radical excision of the affected part of the lung. (1) drainage operations are palliative only and aim either at drainage of abscess cavities or, by making a permanent bronchial fistula, at reducing the amount of spu- tum. (2) all the forms of collapsing operations which are in use for pulmonary tuberculosis have their place in the treatment of bronchiectasis. they are phrenicotomy, pneumothorax, extra- pleural pneumolysis and extrapleural thoracoplasty. the prin- ciples involved are, firstly, that the spaces in which secretions collect and decompose are obliterated by the collapse of the lung, and secondly that the new formed fibrous tissue in its con- traction no longer pulls open the walls of the bronchi softened by inflammation, but can now pull inward the mobilised walls of the thorax. (3) in certain types of bronchiectasis the results of collapsing operations are not satisfactory, and in consequence amputation of the affected lobe or lobes of the lung has been performed. this procedure has produced real cures but the mortality of the operation is high. the longest series of cases is that published by hi. lilienthal, of new york (arch. of surg., heat vol. 8, 1924), whose mortality is nearly 50%. many of his patients were, however, desperately ill. evarts a. graham (arch. of surg., vol. 10, p. 392, 1925) has practised pneumectomy by the actual cautery, in one or several stages, on 20 patients. of these, 50% were cured, 30% improved and 20% died. intrathoracic tumours.—innocent intrathoracic tumours are being diagnosed with greater frequency owing to the increasing use of x-rays. h.c. jacobaeus and e. key (acta. chir. scand, vol. 53, p. 573, 1921) have successfully removed four fibromata by the transpleural route. t. p. dunhill (br. jour. of surg., 1922) removed a fibroma by gask’s modified sternum splitting operation. j. e. h. roberts (tr. roy. soc. med., 1926) has removed four innocent tumours, two fibromas, an encapsuled endothelioma and a dermoid cyst, three by the transpleural route and one by sternum splitting. malignant tumours of the lung —radical operations for the removal of carcinomata of the lung are rarely possible owing to the later stage at which patients come under observation. f. sauerbruch (chir. der brustorgane) has operated in five cases: one, with a carcinoma of the lower lobe the size of a small fist, was alive five years later; another was alive after three years. sauerbruch advises a two-stage operation or three-stage opera- tion, ligature of the branch of the pulmonary artery to the affected lobe being done as a preliminary followed by thora- coplasty and resection of lung. palliative operations often give great relief from distressing symptoms and prolong life for many months; they are (1) drainage of a secondary lung abscess or empyema, (2) drainage of a sterile abscess due to necrosis of growth, (3) exposure of the growth for the insertion of radium, (4) treatment by x-rays. binliography.—f. sauerbruch’s monumental work, chirurgie der brustorgane, 2 vol. (1920 and 1925); l. guibal, trattement chir- urgical de la dilatation bronchique (1924); d. s. allen, “ intracardiac surgery,” arch. surg., vol. 8, 317-25 (jan. 1924); j. alexander, sur- gery of pulmonary tuberculosis (1925); j. gravesen, surgical treat- ment of pulmonary and pleural tuberculosis (1925); d. s. maguire, “ successful cardiorrhaphy,’’ surg., gynec. & obst., 40, 623-5 (may 1925); c. s. beck and r. l. moore, ‘' significance of pericardium in relation to surgery of the heart,” arch. surg., 11, 550-77 (oct. 1925); j. h. long, ‘‘ cardiorrhaphy,” boston mm. and s. j, (dec. 24 1925); h. lilienthal, thoracic surgery, 2 vol. (1925). (ji, ei: fa ra) heat (sce 13.135).—the summary of recent works, given below, is arranged for convenience as far as possible in the order of the earlier articles connected with heat in the 11th ed. of the e.b., as enumerated in 13.157, and references to them are made where necessary. international notation.—the symbolic notation here adopted is based on that recommended by the international commission for the unification of physico-chemical symbols at their meet- ing at brussels in 1913, as extended by a special committee of the physical society of london under the presidency of sir j. j. thomson. fortunately their recommendations coincide in the main with the notation employed in the rith ed. of the #.b., but a few changes have been made for the sake of uniformity, as indicated in the following list. alphabetic index of symbols. a=1/j, reciprocal of mechanical equivalent of heat. a, numerical factor for reducing pv to heat units. 13, constant of integration in expressions for e and h. b, co-volume in characteristic equation of gas. c, cooling-effect of joule and thomson (see 27.901). c, co-aggregation volume in gas-equation. e, intrinsic energy. g, gibbs’ function, t@—tt. i{, total heat of vapour, e+apv. h, total heat of liquid. j, joule’s equivalent. kx, &, thermal conductivity, and diffusivity. l, latent heat. m, mass. m, molecular weight or mass-flow. n, number of atoms or molecules. n, index in formula for c. p, pressure generally. p, saturation-pressure. o, quantity of heat energy. 327 r, gas-constant in pv=rt. s, specific heat of vapour; s, of liquid. t, absolute tempcrature; f, from o°c. u, velocity of motion. v, specific volume of vapour; 2, of liquid. w, work. n, cross-section of pipe or nozzle. @, entropy of vapour; ¢, of liquid. 8, radiation constant in br;t. y, ratio of specific heats of gas, a, velocity of light, 3 x10" cms/sec. a, wave-leneth; », frequency. n, viscosity of gas. calorimetry units of heat.—one of the most fundamental points in the measurement of heat is the relation between the practical units corresponding to the various methods discussed in the earlier article (see 5.60), in which the most important experimental evidence then available was described and reviewed. some of the conclusions reached have since been contested, but additional experimental evidence has been obtained which seems to confirm the views previously maintained. the expcriments of rowland by the mechanical method, agreeing closely with those of joule when reduced to the scale of the gas thermometer, showed that the gram-calorie at 20° c. (defined as the quantity of heat required to raise the temperature of 1 gram of water at 20° c. under atmospheric pressure by 1° c. on the scale of the hydrogen thermometer) was equivalent to 4:180 joules of mechanical energy. ‘those of reynolds and moorby between o and 100° c. gave the equivalent of the gram-calorie as 4-1832 joules for the mean of the whole range, showing that the mean calorie was nearly the same as the calorie at 20° c., in contraclic- tion to the results of earlier experimentalists who had obtained much higher values for the mean calorie. the best of the previous results by the method of mixtures for the variation of the specific heat of water between 0° and 100° c. were those of liidin (see 5.64, fig. 6), which gave a somewhat improbable curve for the variation, indicating a value 4-206 joules for the equivalent of the mean calorie, if the calorie at 20° c. was equivalent to 4-180. most of the older results for the mean calorie, e.g., those of dieteric1 (i red. ann., 33, p. 417, 1888), giving 4-244 by an electrical method with an ice-calorimeter, were much higher than liidin’s. on the other hand, the continuous electrical method (see 5.65), in which platinum thermometers were employed in place of mercury thermometers, while agrecing very closely with rowland’s results from 5° to 30° c., gave a much slower rate of increase than liidin’s for the specific heat between 40° and 100° c., and a value 4-186 joules for the mean caloric, confirming reynolds and moorby. the later experiments of dicterici, by the method of the ice- calorimeter employing a 10 times smaller current with a coil of higher resistance in order to reduce the uncertain errors of the electrical measurement, gave an equivalent 4:192 joules for the mean calorie. he also redetermined the constant of the ice-calorim- eter, using water at 100° c, sealed in thin bulbs of quartz-glass, and obtained a value 15-491 milligrams of mercury per mean calorie, appreciably higher than the value 15-44 previously employed. this has since been confirmed by e. griffiths (proc. phys. sec., 26, p. 1, 1913) who found the value 15-486 for a mean calorie of 4-184 joules. owing to the smallness of the quantities of heat available for measurement at low temperatures, the ice-calorimeter is unsuit- able for investigating the variation of specific heat in the neigh- bourhood of the freezing-point, but the observations of dieterici at temperatures above 100° c. by the same method gave a rate of increase of the specific heat of water slightly exceeding that found by regnault, which could not be reconciled with ltidin’s curve showing a maximum of specific heat at 87° c. messrs. w. r. and w.e. bousfield (pail. trans., a, 211, pp. 199- 251, 1911) succeeded in reproducing liidin’s results with remark- able fidelity by a most ingenious method of electric heating with a vacuum-jacket calorimeter. the heating-coil consisted of a long spiral of small-bore glass tubing filled with mercury, the expansion of which in a capillary tube was made to indicate the actual tem- perature of the mercury at any time when traversed by the electric current. the observers were thus enabled to avoid the source of error due to the superheating of the conductor above the tempcra- ture of the calorimeter. the uncertainty of heat-loss by evapora- tion from the surface of the water was minimised by protecting the surface with a cover in the form of a metal box maintained as nearly as possible at the same temperature as the water during an experi- ment. the rise of temperature over predetermined ranges, 0°13, 13°-27°, etc., was observed with suitable mercury thermometers of limited scale, standardised at the national physical laboratory. the corresponding quantities of electrical energy supplied, when corrected for external heat-loss and for the thermal capacity of the calorimeter, gave the increase of total heat of water, or the mean specific heat over each range. by adding the increments of total 328 heat for each range, the variation of the total heat 4, or the small difference #—#, could be obtained at each of the points of observa- tion, as in the following table:— tempcrature c, os ee ees eee et es bousfield 0:058]0-058/0-059|0-124/0-242/0-306]. ... liidin 0:057|0-059|0-064/0-119/0-285]0-371/0-633 formula (1). 0-070|0-072|0-054|0-038|0-046|0-062|0-159 dieterici 0-o10|0-011|0-013|0-031|0-090/0-128]0-303 bousheld's observations did not extend beyond 80°, owing to the difficulty of excessive evaporation with an open calorimeter. ac- cording to his curves, the corresponding values of the specific heat appear to be approaching a maximum at 80° c., a little lower than that shown by liidin’s curve. the value of the specific heat at 80° c., according to liidin's formula, is 1-0184 in terms of the specific heat at 20° c. taken as unity, and exceeds the value given by the continuous electric method by 1-55°%. this looks alarm- ing at frst sight, but the method of comparison in terms of the actual specific heat, though commonly adopted, is really unfair, because the quantity actually observed in liidin’s method is the total heat, which shows a difference of only 0-31 calorie according to the above table at 80°c. dieterici's observations at 100° c., where they were most reliable, differ by only 0-14 °, from the con- tinuous electrical method, and he cloes not claim an order of accuracy greater than o-1 °% for the ice-calorimcter. continuous mixture method.—since the number of separate deter- minations of the specific heat of water at points between 50° and 100° c. by the continuous electric method was only 12, and since these were made under conditions of exceptional difficulty, and differed most widely from the values found by liidin and bous- field, it was felt to be desirable to confirm the variation in this region by an entirely independent method of equal accuracy. the continuous mixture method (bakerian lecture, phi. trans., a, 1912, vol. 212, pp. 1-32) was devised for this purpose, and consisted in passing a steady current of water, initially at roo’ c., through an interchanger, in which it gave up a large part of its heat to a cur- rent of cold water initially at 25° c., emerging at a temperature in the neighbourhood of 70° c., without having actually mixed with the cold current. the same current was then cooled to an accu- rately regulated temperature in the neighbourhood of 25° c., and re-entered the interchanger as the cold current. the point of the method is that the circulation is continuous, so that the water equivalent of the interchanger is not required, and that the hot and cold currents are the same, so that the quantity of the current divides " out of the equation (except in the small term represeniing the exter- nal heat-loss) and need not be determined with an accuracy greater than 1%, since the external heat-loss can easily be reduced to a small fraction of 1% of the heat-exchange between the currents. the actual temperatures 4 and f of the hot current on entering and leaving the interchanger, and those of the cold current, ¢; and t4, were observed with platinum thermometers to o-oo1° c. if s’ is the mean specific heat of the hot current between 4 and &, and s’’ that of the cold current between e and f;, we have the equation sh — ty) =s'"(h, —ts) +x/m, where x is the external heat-loss in gram-calories per second, and m the value of the water current in grams per second. the heat- loss was determined, as in the continuous electric method, by vary- ing the flow m while keeping the temperatures the same. in a large number of trials it was found that the ratio of s’ to s’’ agreed with the value 1-0050 given by the continuous electric method, but disagreed materially with the value given by liidin’s formula. it was concluded that the discrepancy from liidin’s formula was probably to be attributed to the unavoidable errors of his mcthod, due to losses by evaporation at temperatures above 50°, and to the uncertainties of zero-point and stem-exposure which cannot be eliminated in the employment of mercury thermometers. formulae for the specific heat of water.—it is usual to employ an empirical formula of the type, s=z-+at+df-+ct®+etc., which is familiar and convenient for the application of the method of least squares to the results of observation. the formulae most often quoted for water are those of liidin and dicterici, which are as follows in terms of the calorie at 20° c.:— dicterici, liidin, s=i-0013 —0:0104(¢/100) +0-0208 (t/ro0)? s$ =1i—0-07668(t/100) +0-196(#/100)? —o-116(#/100)3 + 0-00025 0-040 == 0-030 the probable errors of the coefficients, as given by liidin, are shown in the line below his formula. the formula of dieterict repre- sents his observations satisfactorily from 50° to 300° c., but does not apply to the variation near the freezing point, which cannot be represented satisfactorily by this type of formula without an additional term. the formula of liidin is fairly accurate between o° and 25°, but appears to give results about 1°% too high between 60° and go° c. it is also inconvenient in practice, because the coeffi- cients are large and of opposite signs, giving the small variation required as a difference between relatively large terms. in the pre- plea t liminary reduction of the results of the continuous clectric method (b. a. rep., 1899) it appeared that a formula of this type would be unsuitable, and the observations were accordingly represented by three simple formulae for different ranges of temperature between oo . 2 - a ft o° and 200° c., as given in the previous article (see 5.66). these have since been combined into a single equivalent formula, which is more convenient for several purposes:— $s =0.98536 +0-504/(¢ +20) +0-0084 (1/100) +0-009(¢/100)? (1) the value of the constant is adjusted to make s=1 when t=20°, the other terms are small and positive, and can be calculated with sufficient accuracy for all possible purposes by means of a io-in. slide rule. this formula agrees very closely with the table pre- viously given, but represents a later and more accurate reduction. it is of no theorctical significance, and cannot safely be extrapolated much above 100°c., but still agrees very closely with regnault's observations at 160°c. above 100° c. it is better to use the thermo- dynamical formula (see 27.903) suggested by mecf. gray, which agrees very closely with experiment from 40° to 100° c., but does not represent the increase of specific heat with fall of temperature near the freezing point. gray's formula was re-defined by callendar as representing the change of total heat of water under saturation pres- sure, and then agrees very closely with the observations of dicterici at high temperatures, when corrected to give the change of total heat in place of the intrinsic energy. it has a simple theoretical foundation, and greatly simplifies the thermodynamical relations between liquid and vapour. there is good reason to believe (callen- dar, properties of steam, pp. 160, 196) that it continues to hold satisfactorily right up to the critical point, where the specific heat becomes infinite. . by experiments on the supercooled liquid, prof. hf. t. barnes has shown that the increase of specific heat with fall of temperature con- tinues to follow the same curve above and below the freezing point. by very accurate experiments on mercury, using the continuous electric method, he has shown that a diminution of the specific heat with rise of temperature occurs as in the case of water, but persists up to a minimum at 140° c. it appears probable that a similar phenomenon would be found for all liquids at low vapour pressures, but it is masked in the case of volatile liquids by the opposite effect of the vapour-molecules, as represented by the thermodynamical formula. the diminution of the specific heat of water was attributed by h. a. rowland to the presence of a small proportion of solid- molecules in the liquid near the freezing point. the rapid increase of the specific heat of a solid as the fusing-point is approached may similarly be attributed to the presence of a small but rapidly increas- ing proportion of liquid-molecules in the solid. the proportion required in cither case, to explain the diminution of hardness and rigidity of the solid, or the anomalous expansion of water near the freezing point, is small, but cannot be calculated with certainty on account of our imperfect knowledge of molecular forces and dimen- sions. such a theory would be difficult to verify in any case by experiment for the liquid and solid molecules. specific heat of gases and vapours.—the continuous electric method was first applied in the case of steam (see 27.901) and gave results near 100° corroborating regnault's value at higher tempera- tures. the same method was applied to air and con by w. f. g. swann (phil. trans., a, 1910, vol. 210, p. 199), who found results from 2 to 5° higher than those of regnault. swann’s formula has since been verified by holborn and jakob (zeit. ver. deut. ing., 58, p-. 1429, i9f4) anc it is now generally recognised that this method is the most accurate for the determination of the specific heat of any_fluid at constant pressure. swann's values for air at 20° and 100° c. were closely consistent with those of joly at constant volume (see 5.67), and gave a ratio of specific heats very nearly equal to i-40, as required by the kinetic theory for a diatomic gas. they also towel a very small increase with temperature at the rate of only one-half of 1°% for 100° c. his values for co, verified with improved accuracy the rapid increase with temperature found by regnault and wiedemann for this gas, which amounted to 12% for 100°. this inerease of specific heat was not accounted for on the kinetic theory, which required that all the degrees of freedom of a gas molecule should be equally excited, and should contribute constant terms to the specific heat. the apparent discrepancy was explained (b. a. rep., 1908, p. 340) by supposing that a natural frequency of the gas-molecule would be excited by radiation in direct proportion to the intensity of the corresponding frequency at each temperature. it was shown that a natural frequency having a wave-length of the order of 15 microns would be competent to produce the observed effect in the case of cos, contributing, when fully excited, a term r to the specific heat. an attempt was accord- ingly made to investigate the relation between the variation of the specific heat of gases anc the absarption and emission bands in their infra-red spectra. some qualitative agreement was found, but it was very difficult to make quantitative measurements of the kind required, or to frame a consistent theory. for instance, there is a strong band at 4-4-4:5 microns both in the emission and ab- sorption spectra of steam. this band corresponds to the maxi- mum ordinate of the wave-length spectrum of full radiation at a temperature t=647° c., the critical point of water, and appears to be closcly related to other properties of steam. there is no doubt heat that the properties of any substance are intimately related to the natural frequencies of the molecules, but the form of the relation cannot be predicted with certainty; and the quantitative measure- ments are not yet sufficiently exact to distinguish between many possible hypotheses. , the experiments of a. eucken (sitz. akad., berlin, 33.1, p. 141, 1912} on the specific heat of hydrogen at low temperatures were very instructive in this connection. the gas was electrically heated at various temperatures in a thin steel vessel under considerable pressure at constant volume. the specific heat was found to dimin- ish from nearly 5r/2 at ordinary temperatures to nearly 3r/2 at t =60°, after which it remained practically constant down to t =35. the experiments were undoubtedly of considerable difficulty, but there scems no.reason to doubt their substantial accuracy. eucken's results have recently been confirmed with remarkable precision by j. h. brinkworth (proc. rey. soc. (a)207, p. 542, 1925) using an entirely independent method of experiment. he observed the covling effect in adiabatic expansion with a compensated platinum thermometer at various temperatures between 20° c. and —183° €., and deduced the corresponding values of the ratio of the specific heats at constant pressure and at constant volume. the actual specific heats at any temperature could be deduced with certainty from these observations. this method is unaffected by the thermal capacity of the containing vessel, whereas in eucken’s method the thermal capacity of the vessel must be known with consider- able accuracy. brinkworth also showed that the heat-loss could be most satisfac- torily climinated by using vessels of different sizes. assuming that the variation of the specific heat was due to the response of some particular frequency of the molecule to the same frequency in natural radiation at each temperature, he states that callendar's radiation formula fits the observations better than planck’s but that satisfactory agreement cannot be obtained by assuming a single frequency. reiche’s calculations do not seem to improve the agreement. the effect is probably due to an absorption band of \ =25-30 in the infra-red. specific heat of solids at low temperatures.—the early experi- ments of sir j. dewar, sir w. a. tilden and others, had shown that solids at low temperatures deviated from dulong and petit’s law. of the constancy of atomic heat in the same way as carbon, boron and silicon, at ordinary temperatures, but they failed to show the full extent of the deviation, or to indicate a probable explanation. a. einstein suggested (ann. phys., 22, p. 180, 1907) that the atom of a solid might be regarded as an electric resonator with three de- grees of freedom possessing a particular frequency, independent of the temperature, and capable of responding to the same frequency of radiation. adopting planck’s theory and radiation formula, he showed that the specific heat at constant volume should approach the limit 3r =5-94 calorics per gram-atom at high temperatures, as required by dulong and petit’s law, but that the variation at low temperatures should be given by the expression (2) s=3rz%e*/(e* —1)? = 3rf(z) where =f»/t=c/xt, as in planck’s formula. the symbol » de- notes the natural frequency of the atoms, and a the correspond- ing wave-length in cm. such that va=a=3 x10", the velocity of light. the constant, ba=c, is wicn’s constant of radiation. taking h. f. weber’s observations on the variation of the specific heat of the diamond, extending from t=222° to 1258°, einstein showed that they agreed qualitatively with this formula, if we could assume the diamond atoms to possess a single frequency cor- responding to the wave-length 11 microns. taking the substances, cafl, nacl, kcl, caco; and sioz, for which the optical fre- quencies in the infra-red were known, he showed that the frequen- cies agreed in order of magnitude with those required by his formula, but that the observed wave-lengths were somewhat shorter than those calculated from the specific heats. this could be attributed to the fact that most of the substances showed more than one fre- quency, and that the frequencies were not strictly monochromatic, as indicated by the width of the corresponding absorption bands. in any case there were other effects, such as work of expansion, included in the specific heats as ordinarily measured, and it might be doubted whether the optical frequencies corresponded exactly with the thermal vibrations of the atoms. an important series of experimental measurements, extending down to the temperature of liquid hydrogen, was made by w. nernst, f. a. lindemann and their collaborators (sitz, akad., berlin, p. 494, 1911), on a number of metals and other solids, including those for which the optical frequencies were known. they found, as already indicated, that einstein’s formula gave too low values for the specific heats at low temperatures, if the optical frequencies were assumed in calculating the value of f(z), and that much better agreement could be obtained by taking the mean of f(s) for the optical frequency, and a similar term, f(2/2) at half the optical frequency :— s=3r[ fle) +f(e/2)l/2 = 3rf"(e) (3) the same function, f’’(z), of z was assumed to apply to other sub- stances, such as the metals, but .the appropriate values of = were selected to fit the observations on the specific heats. some sub- stances, such as sio, (in the forms of quartz and quartz-glass) and benzine, cshs, which gave a different type of curve, were represented by formulae with two or three different values of 2, each value of f’(s) being multiplied by a fractional coefficient representing the proportion in which the corresponding molecule was supposed to be present. but such cases could not be regarded as a verification of the theory, because it would obviously be possible to represent almost any type of variation in this way. einstein 329 objected that even the simplest of these formulae, namely (3), was too empirical to be satisfactory from a theoretical standpoint; that a cubical crystal, such as kci, or nac}, could not have two different frequencies; and that there was no evidence in either case of an optical frequency with half the experimental value, since, according to rubens, the crystals became again transparent before this frequency was reached, and had a value of the refractive index which was nearly normal. he also indicated two other objections to the ‘“ quantum ” theory on which planck’s formula was based. (1) according to the quantum theory it did not follow, as required by the classical mechanics, that the oscillator with three degrees of freedom would have three times the energy of a linear oscillator. (2) it was very difficult to conceive the distribution of energy among the oscillators at low temperatures required by the theory. thus for the diamond at t =73° only one molecule in 100 millions would possess a single quantum of energy, all the rest would be absolutely quiescent. it was physically impossible to conccive such a distribu- tion of energy, which moreover would make the thermal conductiv- ity of the diamond at such temperatures entirely negligible, whereas, according to eucken, it was nearly as great as that of copper at ordinary temperatures. for these reasons einstein preferred to rely mainly on the expression for the energy of an electric oscillator in equilibrium with radiation as deduced from maxwell's equa- tions, and to regard planck's formula for the distribution of energy in full radiation simply as representing the results of experiment. debye’s theory of specific iteat of solids ——the theory now most commonly accepted is that of p. debye (amn. phys., 39, p. 789, 1912), who attributes the heat energy to mechanical or acoustic vibrations of the solid with all possible frequencies up to a certain limit »,. according to a theorem attributed to the late lord ray- leigh (sound, i., p. 129, 1877) the number of possible degrees of free- dom of a system of n discontinuous mass-points will be 3n. accord- ing to another theorem by the same author (phil. afag., 49, p. 539, 1900), the number of possible frequencies in a given volume of a continuous medium between the limits » and »+dyv may be repre- sented by c’v?dvy, where c’ is a constant depending on the volume and the velocity of propagation. the total number of possible frequencies from o up toa limit vm is c’vn3/3. if we equate this to 3n, we find c'=9n/»,3. adopting planck's expression for the energy of an electric oscillator with one degree of freedom as apply- ing to each possible frequency of the n atoms in a gram-atom, we obtain the energy (rt/n)s/(e7—1) for each frequency. multi- plying this by the number of frequencies between v and »+dp, namely (qn/»,°)dv, and integrating from 0 to ym, we obtain the energy of a gram-atom at t, from which the specific heat at constant volume is obtained by differentiation with regard to t. unfortunately the integral cannot be expressed in finite terms and is too complicated to reproduce here. it is evident, however, that it will be a function of zn, or b¥m/t, or t/t, where tr =8rn. thus the form of the curve representing the variation of the specific heat (which depends on a single parameter i, or y,) is the same for all substances on debvye’s theory, if the temperature scale is altered for each in proportion to »,». this point has been very care- fully tested by e. h. grithths and e. griffiths (phil. trans., a, 214, pp. 319-357) for the metals al, ag, cd, cu, fe, na, pb, zn. their results indicate qualitative agreement with the theory, but show characteristic differences, greatly exceeding the limit of experi- mental error, which may possibly be attributed to other effects not included in the simple theory. thus the curve for ie differs from that for cu by nearly 20%) between corresponding temperatures, which may be attributed to the magnetic properties of fe. the curve for na shows a rapid rise towards the melting point, reach- ing an excess of 25% above 3k, followed by a diminution of specific heat for the liquid, as in the case of watcr and mercury. many simple compounds, such as nac}, show curves of a very similar type to the metals, which has been used as an argument that the specific heat must be attributed entirely to the atoms, and that the free electrons supposed to exist in metals cannot make any appre- ciable contribution. thus if there were two free electrons per atom, as required by some theories, the electrons alone would account for the whole specific heat according to the kinetic theory at ordi- nary temperatures; and it would be necessary to suppose that the number of free electrons diminished to zero at low temperatures, which would make it difficult to account for the enormous increase in electric conductivity of pure metals demonstrated by kamer- lingh onnes in the neighbourhood of the absolute zero. one of the commonest objections to debye’s theory is the arbi- trary nature of the assumption of an abrupt limit of frequency ym. this assumption is made on account of its simplicity, but is highly improbable from a physical standpoint, though it might be expected to give results of the right order of magnitude. w. sutherland (phi. mag., 20, p. 657, 1910) had previously shown that the wave-length of the elastic vibrations of solids was of the same order of magni- tude as the distance between the atoms for frequencies correspond- ing to the optical frequencies in the infra-red, so far as these were known. if the forces holding the atoms in place in a crystal lattice are electromagnetic, as commonly assumed, we should expect that the energy would be shared between matter and aether, and that the natural frequencies of the optical and mechanical vibrations would be the same. the wave-length and velocity of the natural 330 frequency as measured outside the crystal would be reduced inside the crystal in the same proportion as the ratio of the velocity of light to that of an elastic vibration, or of the wave-length outside the crystal to the lattice constant, 7.e., in the case of rocksalt, nacl, about in the ratio 2x10° to i. since the energy in the cube of the wave-length remains constant, the encrgy-density of the external radiation of the natural frequency would be increased in the cube of this ratio, and would be of the right order of magnitude to explain the specific heat of the solid on the usual theory of resonance as applied by einstein. we have seen, however, that the assumption of planck's radiation formula gives too low a value for the specific heat at low temperatures on einstein’s theory. if on the other hand we interpret lord rayleigh’s formula, namely c’te7**d», as representing the partial pressure pdv of radiation between the limits of frequency » and v+dy, the latent heat of emission or ab- sorption of radiation per unit volume between the same limits, according to carnot’s principle, is represented by the expression (4) t(dp/dt) =c’t + 2)e7 and the total heat of a gram-atom of solid in equilibrium with radia- tion having this distribution of energy is given by h=3rt(1+2)e~ (5) the specific heat as ordinarily measured, when the external pres- sure is small as compared with internal pressure, will be simply s=3ru+24+2)etm | (6) this expression, unlike that similarly deduced from planck’s for- mula, gives good agreement with the observed value of the specific heat in the case of rocksalt, when the optical frequency corre- sponding to 5£ microns is assumed, at a temperature corresponding to the maximum of the frequency curve, where z=2-732, t=100°, and s=8-67 (doubled for a gram-molecule of nacl). we should ex- pect to get good agreement at this point, in spite of the fact that the actual vibrations in a solid cannot be strictly monochromatic (as einstein pointed out) but extend for a distance of an octave or more on either side of the maximum, as indicated by the absorption spectrum. the effect of this is to reduce the steepness of the mono- chromatic curve, bringing it into goo agreement with observation at high and low temperatures, without materially affecting the agreement at the mean point corresponding to the maximum of the frequency curve. if we assume the value 4,,t =o-290 for the wave-length am (corresponding to the maximum ordinate of the wave-length spectrum of full radiation at t), in deducing the appropriate value of wien’s constant 8a in formula (4), the maxi- mum ordinate comes out the same as in planck's formula, provided that the same value of the stefan-boltzmann constant o is assumed in the fourth power law ¢t! for the total radiation. the two curves also agree so closely throughout their whole extent that it would be very difficult to decide between them by experiments on radiation. we should therefore be justified, according to einstein's reasoning, in applying formula (4) in the deduction of the specific heat of a solid, especially when we find that the result gives such good qualitative agreement with the optical frequencies. an obvious objection to debye's theory in the case of transparent substances, such as quartz sad rocksalt, is that, if the atoms have all possible frequencies below a certain limit, they ought to be com- pletely opaque in this region, and to become suddenly transparent when the limit », is surpassed. experiment shows, however, that, e.g., quartz, which begins to be opaque at about four microns, and has optical frequencies corresponding to 9 and 21 microns approximately, and possibly one lower, becomes almost perfectly transparent below 100 microns. the variation of its specific heat is of an entirely different type to that given by debye’s theory but corresponds closely, according to formula (4), with its optical fre- quencies, ice and benzol, which are also hexagonal, show a varia- tion of specific heat similar to quartz, according to sir j. dewar. the corresponding optical frequencies have not yet been observed, but it appears that water must have some frequencies below 100 microns to account for its remarkable opacity to long wave-lengths, and the variation of its specific heat. we should naturally expect that the torsional vibrations of an elastic solid, which are of the same kind as those of light, would be excited by radiation, and would be intimately connected with the optical frequencics. it is quite possible, however, that the compressional vibrations, which are of a different type, and propagated with a different velocity (that of sound), would continue to exist at low tempcratures with- out affecting the transparency. these acoustic vibrations, though not capable of being excited directly by radiation, would be neces- sarily excited by the impacts of the molecules of the surrounding gas, with a distribution of energy corresponding to the maxwellian law, and might be expected to provide a term in the specific heat of a somewhat similar character to the debye term for comprcs- sional waves at low tempcratures. it is noteworthy that nernst and lindemann in their latest reductions have found it necessary to retain the original einstein term f(z) for transparent substances in their formula (3), but have replaced the hypothetical term f(z/2) by aterm of the debye type. the appropriate frequencics are cal- culated in most cases by lindemann’s semi-empirical formula from the molecular weight 7, the atomic volume v, and the tempcrature heat of fusion ty, but with different values of the constants for the two terms, as follows:— ye = 2-12 x10°(ty/m2) 4v-75 pm = 3°08 x 10!2(ty/i) 6 v4 (7) of which the first gives the optical frequency of einstein and the second that of debye. the cube root of the atomic volume is pro- portional to the lattice constant, and the clastic constants of a solid must be closely related to the temperature of fusion. nernst and lindemann assign equal importance to the two terms, but we should naturally expect from elastic theory, as given by debye and other previous writers, that the numcrical coefficients should have differ- ent values, and should be proportional to 1/4? for the compres- sional waves, where 7 is the velocity of sound, and 2/1.? for the tor- sional waves, where #2 is the velocity of light 77 the solid for the particular optical frequency considered. this may not fit so well with planck’s radiation formula for the einstein term, but appears to give better agreement with experiment if formula (4) is substi- tuted for planck's. the appropriate frequencies cannot be calcu- lated from the elastic constants for a discontinuous medium with- out introducing arbitrary hypotheses, which are unsatisfactory, because the effect of the hypothesis selected is most important at the point where the discontinuity commences and it is difficult to avoid selecting an hypothesis to give the desired result. there is the further difficulty that the values of the elastic constants are somewhat uncertain and liable to vary with temperature and to depend on the particular specimen tested, especially with metals. sir j. dewar (prac. r. s., 1913, a, 89, pp. 158-169) has measured the mean specific heats of the elements between the boiling points of hydrogen and nitrogen by means of his liquid hydrogen calorim- eter. the results for the specific heats, when plotted against the atomic weights, give a curve showing a most remarkable coinci- dence with the well-known curve of atomic volume as a periodic function of the atomic weight. in other words, the specific heat is nearly proportional to the atomic volume, or to the cube of the lattice constant, for similar substances, at this low temperature, corresponding to a mean about t=50°. the relation does not pre- tend to be exact, though it is a fair approximation over the range 20° to 80°, but it illustrates the point that the atomic volume is the most important factor in determining the frequencies. in the case of the metals, which are opaque to all frequencies below a certain limit, we should expect the possible frequencies to extend over a considerable range, and to be grouped about a mean in a similar way to the velocities of gas molecules on the kinetic theory, but there are many possible alternatives to the somewhat arbitrary hypothesis of debye. we might suppose, for instance, ihat of n molecules in a gram-molecule, the number possessing the frequencies between the hmits v and »+dv was represented by an expression of the type (n/2)e7*x2dx (8) in which x=vfvp9=8v/to=92, where z denotes bvy/t, and @6=t/to. multiplying this by expression (6) divided by n for the specific heat of a single molecule of frequency », at a temperature t, and inte- grating the product from 0 to ©, we obtain for the specific heat of a gram-molecule (9) 5=3r(69/(1 +8)*) (1 +3/(1 +6) +12/(1 +8?) this is much simpler than debye’s expression, but gives a very similar curve. the mean frequency, ».=3%, is nearly the same as debye’s limiting frequency. more accurately, debye’s character- istic temperature corresponds to 2-917t9, in place of 3ts, on account of the difference in the values of the constant 8, which are in the ratio 4-9651/4-8284 in planck’s and rayleigh’s formulae for radia- tion. if debye’s scale is multiplied by 2-91, his curve agrees very closely with (9) from 6=0-6 to @=1-0. below 6=0-6, (9) agrees better with the nernst-lindemann curve (3), except that (9g) tends to vary as t* at very low tempcraturcs, instead of vanishing expo- nentially. above @=1, the curve (9) lies above debye's by a quan- tity corresponding to the difference of the specific heats at constant pressure and volume. this is to be expected, because (9) represents the rate of change of total heat, which is the same as that of intrin- sic energy for all practical purposes under the condition of small external pressure and negligible expansion. thus in the case of water under atmospheric pressure, the increase of total heat between o° and 100° c. is 100 cals. c., and exceeds that of intrinsic energy between the same limits by only o-oor cal. c., which is 100 times smaller than the limit of accuracy of observation; whereas the change of total heat at constant volume between the same limits in the case of water exceeds that of intrinsic energy by 21 cals., approxi- mately; but the correction from constant volume to constant pres- sure is very uncertain, even in the best known cases. it therefore appears to be more logical to employ a formula giving the specific heat at constant pressure directly, in place of applying an uncer- tain correction. it should be observed, however, that (9) assumes the mean frequency vy, to be independent of t, as in dechbye’'s formula, which may he a good approximation in many cases, but cannot be exactly true if the molecule changes its state. curve (9) reaches s=3r a little above @=2, attains a maximum 3:195r at 6=4, and falls again to 3r at @=2. the fall is of the right heat order of magnitude to explain the diminution of specific heat in the case of water, mercury and sodium. the distribution postulated in (8) appears 10 apply fairly to most of the metals, but it fails notably for many other substances. such cases might be treated empirically by modifying the distribution, or assuming special frequencies, but such hypotheses would be of little value unless their physical meaning could be interpreted with reference to other properties of the substances. conduction of heat in roro the very attractive theories of p. drude and h. a. lorentz were still commonly maintained, and were continually being applied to the explanation of electrical and thermal effects. according to their views a metal contained a number of free electrons moving in all directions with velocities corresponding to those of gas-molecules on the kinetic theory. drude showed that this assumption led to an approximately correct value of the ratio of the thermal to the electric conductivity in the case of pure metals, and lorentz showed that it accounted for the long wave radiation from hot bodies. there were numerous other applications of the theory which appeared to correspond in a remarkable manner with experimental facts, but there were also serious difficulties which appeared to render the adoption of such a theory premature. the fluid state of scientific opinion on the subject in 1911 is well illustrated by the views expressed about that time by j. h. jeans. in the report of the solvay congress, 1911, on the theory of radiation and quanta (gauthier villars, paris, 1912), assuming that there were two free electrons per atom of the metal, jeans took the view that the specific heat of metals was entirely due to the movement of free electrons and not at all to the movements of the atoms, “ a hypothesis which accords well with our know]- edge of the internal movements of solids.”? on the other hand, in his report on the quantum theory (phys. soc., london, 1914), he adopted the theory of debye (according to which the specific heat was entirely due to the movements of the atoms) as prob- ably “ destined to be final,’ and concluded that the free elec- trons do not contribute sensibly to the specific heat. sir j. j. thomson, corpuscular theory of matter (1907), had already pointed out that the number of free electrons required to explain thermal and electric conductivity was too large to reconcile with the facts of specific heat on the assumption that the electrons possessed the same energy of agitation as gas mole- cules at the same temperature, and had proposed an alternative theory (oc. cit., p. 86) previously suggested in his applications of dynamics to physics and chemisiry (1888). according to this view, the metallic atoms, owing to their close proximity in the solid state, were capable, under the influence of an electric field, of forming grotthus chains, along which they could exchange electrons. there were no free electrons in the sense contem- plated by drude and lorentz, with velocities depending on the temperature and contributing to the specific heat, but the ther- mal agitation of the atoms tended to break up the chains, so that their number and length varied with the electric field in the manner required to explain the relation between electric and thermal conductivity and many other effects. in a later paper (proc. phys. soc., 27, p. 527, 1915), the same theory was applied to explain the striking phenomena of super- conductivity discovered by onnes, who found that at very low temperatures, in perfectly pure metals, a current once started might continue for days instead of stopping almost instantane- ously on the cessation of the exciting field. according to j. j. thomson’s theory, it would naturally follow that, below a certain point, the thermal agitation would be insufficient to break up the chains when once they were formed, which would explain why it is that the electric resistance of most pure mctals tends to vanish (apart from impurities) at a temperature above the absolute zero. a working hypothesis of this kind is very useful to the experimentalist as affording a mental picture of the physi- cal conditions, and may help to explain the remaining difli- culties with regard to the specific heats. conductivity of gases.—prof. knudsen drew special attention (solvay report, p. 133) to the data for the thermal conductivity of gases, as being more scarce and discordant, owing to experimental 331 difficulties, than determinations of other properties, and as requiring attentive examination for the elucidation of the law of action be- tween molecules. the hot-wire method of t. andrews (phil. trans., 1840) offers special facilities for relative measurements, such as the comparison of conductivitics of different gases, or of the same gas at different temperatures, and has frequently been applied with this object in recent years. it has also been improved by introducing the usual compensation for end-cffects, and employing more accurate methods of electrical measurement. but it remains liable to the difficulties depending on the small dimensions of the wire, and the uncertainty of the corrections for convection and radiation. for these reasons the parallel plate method, adopted by ef. o. ilercus and t. h. laby (proc. r. s., a, 95, p. 190, 1918) for measuring the absolute conductivity of air, deserves special mention, owing to the great care with which the method was applied, and the complete elimination of convection effects. they also give a very complete reduction of previous results for different gases with the view of testing the value of the numerical coefficient f in the relation, k =fns, between the conductivity &, the viscosity 7, and the specific heat s at constant volume. according to the theoretical investiga- tions of s. chapman (phil. trans., a, 21, p. 433, 1911) the value of the coefficient f should be 2-5 for a gas constituted of spherically symmetrical molecules, which agrees with maxwell's theory based on the inverse fifth-power law of force, and also with experiment for monatomic molecules. unfortunately the variation of viscosity with temperature does not satisfy the fifth-power law, which re- quires that the viscosity should be directly proportional to t. the conclusion is that monatomic gases may have spherically symmetrical molecules, but that the law of force is different. theory gives no clear indication with regard to the appropriate value of f for other types of molecules. experiment gives approximately a linear relation, f=2-816y—2-2, between f and the ratio of the specific heats. this gives f=7/4 for diatomic gases, which show fair agreement with each other. the experimental values for poly- atomic gases are much less certain. thermodynamics since the general principles of thermodynamics have not undergone any material change for the last 50 years, it will readily be understood that such progress as there is to record relates chiefly to matters of expression or convention, and to the practical application of the principles to engineering prob- lems. the evolution of the steam turbine and the internal com- bustion engine, along thermodynamical lines, has illustrated the importance of an exact and consistent theory of the conditions limiting the efficiency, and of an accurate experimental study of the properties of the working fluid in either case. thus the improvement of the internal-combustion engine has depended greatly on the extension of the thermodynamical efficiency of the cycle by using higher compression-ratios, which has neces- sitated careful attention to the reduction of heat-losses, to the properties of various fuels in respect of detonation, and to the specific heats of the products of combustion at high temperatures. the displacement of the reciprocating engine by the turbine for large power units has similarly depended on the possibility of improving the economy by utilising high vacua. the high speed of the turbine has directed special attention to the impor- tance of losses due to friction and supersaturation, which depend on the rapidity of expansion. the turbine realises the ideal con- dition of steady flow with an exactitude unattainable by the reciprocating engine. this has made it worth while for engineers to adopt the thermodynamical definition of total heat first pro- posed by callendar in the roth ed. of the z.b., in place of reg- nault’s definition, which had sufficed for many years, but con- tinually gave rise to minor difficulties and complications when applied to the turbine. in the article cited, and as repeated in the 11th ed. (see 26.811), the total heat was defined as the ther- modynamic function e+ pv, and was denoted by thespecialsym- bol f in order to distinguish it from regnault’s total heat h, representing the quantity of heat added to the fluid under the condition of constant pressure equal to that of vaporisation. by genera] convention, the symbo] h has now been defined as representing e+pv, a property of the substance depending only on the state, and the symbol q has been allocated to any quan- tity of heat added under special conditions. f-quations of steady flow.—these depend on the law of conserva~- tion of mass, and on the law of conservation of energy, of which thev afford some of the simplest possible illustrations. [hf a fluid is flowing steadily at a constant rate m (mass per second) through a 332 circuit (pipe or nozzle) of variable cross-section x, at a point where the mean volume is v per unit mass, and the mean velocity u units of length per second, we have mv =&ux, where the constant & is unity in any consistent system of units, e.g., if u, ® and v are measured in ft., sq. ft. and cu. ft. respectively. it 1s common prac- tice, however, to measure x in sq. in., which must be reduced to sq. ft. by putting k=1/144; and similarly for other arbitrary sys- tems. if we consider any two points (1 and 2) of a circuit for which m and x are known, the relation mv=kux makes it possible to determine either u or v at each point if the other is known. a sec- onu relation is obtained from the conservation of energy. suppose for example that the points 1 and 2 represent the admission and exhaust of a turbine. when the flow is steady, for each unit mass entering at i, unit mass must leave at 2. unit mass entering at i carries with it its intrinsic energy e,; and its kinetic energy u\?/2g, in addition to which work p,v, is done by the pressure p, in forcing the volume v; into the turbine. reducing these to heat units by the appropriate numerical factors, @ and j, we have finally for the total energy entering the turbine with each unit mass of fluid, h,+u,2/2jz, where h, is the initial value of the total heat, which is always tabulated in heat units per unit mass. similarly the total energy carried out per unit mass at 2 is h,+ u.2/2j}g. since the total quantity of energy existing in the turbine remains constant when the conditions are steady, the excess of the energy carried in over that carried out must be equal to the external work w/j done by the turbine together with the external heat-loss q, both expressed in thermal units per unit mass passing through the turbine. we thus obtain the general equation representing the conservation of energy :— heat-drop, hi—h:=w/j+0+(u2—u,*)/2jg (10) the reduction factors, a, j, g, can be omitted for absolute or c.g.s. units, but it is better to retain them explicitly, because the various quantities can seldom or never be measured in absolute units in practical work and the retention of the symbols saves much trouble and many mistakes. in this equation, as applied to a turbine, the term w/j, represent- ing the external work, is the most important on the right-hand side. the external heat-loss q, and the leaving-loss, depending on the kinetic energy wasted in the exhaust, can be reduced to small cor- rections, which are readily applied. the external work is the equiva- lent of the corrected heat-drop, which can be calculated if the initial and final states of the steam are known. the equation takes exact account of any work wasted in internal friction, which does not appear explicitly in the equation because it affects both sides equally. the same equation can be applied to a reciprocating engine, or to any appliance admitting of steady flow. joule and thomson (phil. trans., 1854-62; proc. r. s., 1856) were the first to employ the function e+pv in their experiments on the flow through a porous plug or orifice. they discussed the various terms in the equation with great precision, but did not apply it to a steam engine, which was first done by hirn and rankine, though the equation is commonly attributed to zeuncr. in an ideal throitling experiment, such as that designed by joule and thomson, the equa- tion shows that the total heat remains constant, i1;=il», pro- vided that ui=u2 and that w and q are negligible. the lines of constant total heat on the pt diagram can be determined by observing the initial and final values of p and t in a sufficient num- ber of throttling experiments. it is then possible to deduce the actual values of eh] under any conditions by measuring the specific heat and latent heat at any one pressure, preferably atmospheric for most fluids. in applying the equation to the discharge through an orifice joule and thomson showed that the kinetic energy gencrated was. equivalent to the drop of e+pv, or h, which follows immediately from equation (10) if w and q are negligible. in the usual case, starting from rest, u,? is negligible as compared with u,?, so that u; is given by the simple relation us=(2jg) (h,—h) (11) for given conditions, v2 is known in terms of hz and ps, so that the discharge m/x per unit area can be deduced by applying the relation m/x=&u/v. joule and thomson showed that the dis- charge would reach a maximum in the case of air under adiabatic conditions when the final pressure after passing the orifice was 0-52 of the initial pressure, a result which had previously been deduced in a similar way by de st. venant and wantzel (comptes rendus, 1839) from poisson's equation for the adiabatic, namely pv*¥=constant. they also showed that the velocity of the dis- charge under this condition was simply related to the velocity of sound in the air at the original temperature and pressure, but they failed to interpret the relation. osborne reynolds (phi. aag., 1886, p. 194), using the same equations for a perfect gas, showed that the velocity at the throat or minimum area of the stream was the same as that of sound in the gas under the same conditions, so that, when this velocity was reached, no further lowering of pressure beyond the throat could possibly increase the discharge. the same result is easily shown to apply to any fluid, either liquid or gas, in the absence of friction. the condition that m is to be a maximum for a given value of x gives d(m/x) =o, whence du/dv = heat u:v. eliminating du/dv by differentiating (11), we obtain, for isentropic flow (& const.) u2 =a]gv(dh/dv)o = ajev2(dp/dv)¢ (12) which is the expression for the velocity of sound. this equation also gives the maximum discharge by substituting m/x for ku/v. in steady-flow calorimetry the drop of 11 between given initial and final states can be deduced from equation (10) by observing the quantity of heat q which must be abstracted, under condt- tions such that w and u? are negligible. the pressure is usually constant, but if there is a large drop of pressure between the initial and final states, as in regnault’s experiments oh the total heat of water, the difficulty is avoided, without changing mi, by using a throttle, which is precisely what regnault did, though he was unable, owing to the defective state of thermodynamics at that time (1847), to appreciate the exact effect of this proceeding. the same method can be applied for measuring the total heat of steam in any state, including the latent heat. in all cases of steady flow the quantity measured is the change of total heat, which is the most important property to determine for steam engines or refrigerating machines working on any modifications of the rankine cycle. on the other hand the intrinsic energy e is the property required for the constant volume cycle of the internal-combustion type. a very simple and instructive illustration of the equation of steady flow is that of the temperature gradient in a fluid under gravity. if a current of air is flowing steadily upwards at a moder- ate speed, the external heat-loss q and the change of kinetic energy are negligible, and the drop of total heat is equivalent to the work done against gravity, giving w/j =1 calorie c. for each 1,400 ft. of ascent. this would evidently be the same for any fluid what- ever. in the case of dry air the specific heat is nearly independent of the temperature and pressure, and the change of h is equal to s(t, —f2), where s=o-241 is the specific heat at constant pressure. the drop of temperature will therefore be 1/0-241 =4-15° c. in 1,400 ft.: or the temperature gradient, 0:296°c. per 100 ft. this result is evidently quite independent of the initial temperature, or pressure, or height, so long as we can afford to neglect the small variations of sand g. in an ascending column of damp air, condensation sets in with formation of cloud as soon as the temperature falls below the dew point. the drop of h remains 1 calorie per 1,400 ft., but the temperature gradient is greatly reduced by the liberation of the jatent heat of the vapour. on the other hand, in a descencling cur- rent, as in the ventilating shaft of a mine, the temperature increases with depth at the rate of nearly 3° c. per 1,000 ft., which, however, is usually much less than the natural gradient of underground tem- perature (due to outflow of heat through the earth’s crust), which sometimes exceeds 10° c. in 1,000 ft. in this case there will be no condensation, but the air may be cooled by evaporation, if the mine is kept wet to reduce dust, as is usually the case. according to equation (10) the rate of increase of tempcrature with depth, denoted by dtjdx, is equal to 1/js, and is uniform in adiabatic flow if s is constant. the pressure gradient, dp/dx, in gravitational units, is equal to the density 1/v, or p/jrt, if r is expressed like s in calories per 1°. divicling by dtjdx, we have dpidt=spirt, giving the adiabatic equation, which is commonly assumed as the starting point to find the temperature gradient. but the reverse order is more instructive as showing why the tem- perature gradient de/dx is uniform, properties of radiation.—the flow of heat by radiation from one body to another at a lower temperature is the commonest case of steady flow. owing to the high velocity of radiation and the absence of thermal capacity in the circuit, the steady state is established in a small fraction of a second if the temperatures of source and sink are constant. the quantity measured in a radiation experiment is not the energy e of the radiation, as is frequently assumed, but the total heat e-- pv, which is the same in the case of radiation as the latent heat of emission, namely vt (dp/dt), for a volume v, according to carnot’s principle. this is universally admitted in the deduction of the fourth-power law (see 13.155), which follows from the fact that the pressure of full radiation is one third of the energy- density, so that the latent heat of emission per unit volume is four times the pressure. the quantity directly measured in experiments on full radiation is the quantity of heat emitted per sq. cm. per second from a black-body or perfect radiator at a temperature t, and is denoted by oi‘, where o is the stefan-boltzmann constant of full radiation. by the geometrical conditions of the problem, the quantity ot! is a‘y times the latent heat per unit volume, or a/3 times the energy-density in an isothermal enclosure at t, where a is the velocity of light. the qualitative verification of the fourth- power law requires only a receiver capable of giving correct relative values of the radiation received, and is now generally accepted as satisfactory; but the absolute measurement of the value of the con- stant « is a much more difficult problem, which has frequently been attacked in recent years without obtaining ,so high a degree of concordance as is desirable in so fundamental a research. ‘the value 5-32 1075 ergs per sq. cm. per second, found by f. kurl baum in 1898 (see 13.155), was accepted for several years, though it rested on a somewhat doubtful value of the absorption coefficient of the bolometer. moreover, the assumption that the radiant energy measured was equivalent to the electric energy required to heat produce the same rise of temperature in the bolometer, was rendered somewhat uncertain by conduction effects at the ends of the strips. a similar bolometer, with the end-effects compensated, as employed in the solar eclipse of 1905, gave the somewhat higher value 5-60 10°, kurlbaum (1912) gave the corrected result 5-45 x1075, f. paschen and w. gerlach, by a modification of angstrem’s method (ann. phys., 38, p. 41, 1912), found the value 5:80 x 1075, which was confirmed by g. a. shakespear (proc. r. s., a, 86, p. 180, 1912), and by h. b. keene (proc. r. s., a, 88, p. 49, 1913), who found 5-67 x 10, and 5-89 x 1075, respectively. w. coblentz (u.s. bur. s#e., 12, pp. 553, 1916), by a method similar to that of paschen and gerlach, found the value 5-72 x 1075, which is a fair mean of the previous results. one of the most promising methods is that of the radio- balance (proc. phys. soc., 23, pp. i-34, 1910), in which radiation received through a measured aperture is completely absorbed in a small copper cup, and is compensated by the peltier cooling-effect due to a current through a thermojunction. unfortunately, these experiments were interrupted by the war, and the final reductions have not yet been completed. there seems to be little doubt that kurlbaum’s original value was too low, but there are many pit- falls in such difficult experiments, and most of the methods adopted are liable to some objections. it is generally admitted that the distribution of energy in the spectrum may he represented within the limits of experimental error by planck’s formula (see 13.156), namely, edd =c'n-8dd/ (e°’ at — 1) (13) if this formula is integrated from 0 to ©, and equated to 6t'4, assuming that it represents the distribution of energy in the spec- trum as observed experimentally, we find for the constant c’, in terms of ¢’ and o, c’=15¢(c';r)*. the value of the distribution constant c’ is most readily deduced from the wave-length a», cor- responding to the maximum ordinate of the energy curve at t, since by wien’s law the product a,,t is the same for all tempera- tures. according to planck’s formula the maximum occurs at the point a,t=c'/4-9651. planck took a,.t=0-294, and «=5-30x 10, giving c’=3-735 x10, and ¢’=1-460. but if a»t =0-289, and ¢ =5-72 x10, then c’ =3-708 x10 and ¢’ = 1-435, according to the latest values of a,,t and o. ‘a comparatively small error in c’, which is raised to the fourth power, suffices to neutralize the error ino. the weak point of the method is that the position of the max- imum of an experimental curve cannot be fixed with any certainty when the curve (as in this case) is far from symmetrical on either side of the maximum. it is too commonly assumed that planck’s radiation formula, in spite of the weighty objections that have repeatedly been urged against it, is so firmly founded in theory and experiment, that no other formula is worth considering in comparison with it. it is also {frequently asserted that no formula based on the “ classical ”’ mechanics can possibly satisfy the required conditions. the argu- ment is somewhat as follows. the number of possible vibrations per unit volume of a continuous medium possessing the properties of the ether, between the limits % and a+da of wave-length, should be represented by 8aa~*dd, according to lord rayleigh’s method of calculation (pil. mag., 49, p. 539, 1900), if the length of path between each reflection is restricted to an integral multiple of half a wave-length. if the different frequencies are regarded as separate inconvertible entities, like the molecules of different gases, between which the energy must, be equally divided, the whole of the energy would accumulate in the infinitely short waves, which is absurd and contradicts experiment, it would be more natural, however, from a physical standpoint to regard lord rayleigh’s formula (8art/n)ew"’’ ata a (14) as corresponding to the partition of energy among a number of similar molecules, according to maxwell's law, which is universally admitted in the kinetic theory of gases, as resulting from the steady state produced by collisions. the steady distribution of energy of radiaiion in equilibrium with matter arises in a similar manner from the deppler effect, by which the energy of a group of waves is changed in the same proportion as the frequency at each encounter with a moving obstacle. the frequency, or the reciprocal of the wave-length, corresponds to the energy, and occurs in much the | same way in rayleigh’s formula, as the square of the velocity, or the kinetic energy, in maxwell’s law. on this view, lord rayleigh’s formula evidently represents the distribution of pressure-energy between the different wave-lengths about a mean value rt/n, which, according to the law of equipartition, should be the same as the pressure-energy of a single gas-molecule at the same temperature. if we take raylcigh’s formula as representing the pressure dis- tribution in full radiation, the expression for the latent heat of absorption l as measured expcrimentally (corresponding to (4) above, but expressed in terms of the wave-length a in the normal spectrum) may be written ldk =c'(t he fyyn ee" at ax (15) integrating from 0 to © we find c”=oc"’’s, the maximum of this curve occurs at the point where c”’/at=2+2¥2, whence c= 4:8284at. the absolute value of the maximum ordinate comes out 333 0:65603(¢t4/a,,). the value of the same ordinate, calculated in the same way for planck’s formula (13), but with c’=4-9651a4,.t comes out 0-65755(¢1t%a,,). it is a curious and significant fact that the maxima should be so nearly the same when the same values of the experimental data are assumed for both curves. the total areas of the two curves are the same, and they agree so closely throughout their whole extent that it would be practically impos- sible to distinguish between them with certainty by experiments on the distribution of heat in the spectrum. the greatest dilfer- ence amounts to about 1% of the maximum ordinate, and occurs near the point a=a,,/2 on the short wave-length side, where the curve is very stecp. this difference becomes quite appreciable in the specific heats, when the curves are differentiated, and secms to jead to better agreement with experiment than planck’s formula as explained above. the most serious difficulty from an experimental standpoint in applying planck's formula, is that the latent heat of emission per unit volume is always tacitly assumed (following planck) to be the same as the energy-density, without taking any account of the pres- sure, whereas the existence of the radiation pressure is universally admitted as the basis of the deduction of the fourth-power law. the work done by the pressure, if it exists, cannot consistently be neglected in experimental measurements of radiation in steady flow. this is one of the most fundamental points in practical thermo- dynamics, but had not up to 1921 received sufficient attention from the mathematicians who have worked so elaborately on the theory. vaporisation a good deal of attention has been devoted in recent years to the study of the properties of vapours employed in heat engines and refrigerating machines. the importance of the thermo- dynamical aspect of the problem has been widely recognized by engineers as the only sure guide to improvements in efficiency, and it has been realised that equations employed to represent the properties of the working fluid must be exactly consistent with the laws of thermodynamics, if it is desired to avoid dis- crepancies in the results of calculations by different methods. the principal properties of vapours were discussed from this point of view in the earlier article (see 27.897). the theory there given still holds good, but it will be of interest to discuss some of the evidence which has since accumulated on the experimental side. the case of steam, for which the experimental data are more accurate than for any other substance, will be taken, as being far the most important to engineers, and as illustrating the properties of vapours at moderate pressures. at high pres- sures, on the other hand, in the neighbourhood of the critical point, the data for steam are somewhat deficient, owing to the difficulty of the experiments, and the tmpracticability of using steam as a working fluid under these conditions. in the critical region the properties of carbonic acid have been most widely studied on account of its use for refrigeration. properties of sieamt.—the equations for steam, first proposed by callendar in the roth ed. of the £.b. (1902), were founded on experi- mental measurements (1) of the specific heats, s and s, of water and steam by the continuous electric method, (2) of the joule- thomson cooling-cffect c with a differential throttling calorimeter and (3) on the adiabatic index y for dry steam with a very sensitive platinum thermometer. these experiments, taken in conjunction with the laws of thermodynamics, sufficed to determine all the re- quired properties fairly accuratcly at moderate pressures. the experiments on the specific heat of water extended from 0° to 100°c., and, when taken in conjunction with those of regnault at higher temperatures, showed that the total heat / under satura- tion pressure could be represented, with sufficient accuracy for the purpose, by the thermodynamic formula h=st+avt (dp/dt) =st4+o0l/(v.—1) (16) | in which the constant s=0-99666 is chosen to make hf at 100°c, = 100 cals. c., or 180 b.t.u. per ib. at 212°f., reckoned from 32°f, the symbol a is the factor for reducing any product of dimensions pv to heat units. when # is in ib. per sq. in. and v in cu. ft. per lb., the reciprocal 1/a (which it is most convenient to use with a slide rule) has the value 9-722 on the centigrade scale, and 5-401 on the fahrenheit scale of temperature. v, and v are the volumes of the dry saturated vapour and the liquid respectively, and dp/dt is rate of increase of saturation pressure p with temperature. when taken in conjunction with clapeyron’s equation for the latent heat, formula (16) gives a very useful relation between the total heat h and the volume v for wet saturated steam in any state, h —st=avt (dp/dt) =pv/ti (17) the factor 1=p/at(dp/dt), which varies slowly and is inde- pendent of the wetness, has been tabulated, as affording the most 334 expeditious and accurate method of cafculating either h or v when the other is known. the relation between h and v when # is given is that most commonly required in practical work. the same formula leads to a simple expression for the entropy ®, h=s log, (t/to) +av(dp/dt) {18) which applies to wet steam of volume v, and also to the liquid if 2 is substituted for v. to represents the freezing point, 273-1°c. or 491-6° f. values ranging from 0-305 at o°c. to 0-665 at 160°c. had been proposed by various writers in 1900 for the specific heat of steam, but the direct measurements by the continuous electric method at at- mospheric pressure from 100° to 160° c, gave results but slightly ex- ceeding those of regnault over the range 124° to 224° c., and showed that the limiting value se at zero pressure was probably nearly constant and equal to 0-477. this was confirmed by l. holborn and h. henning (aun. phys., 18, p. 739, 1905) in a qualitative manner by comparison with air over the range 110° to 820°c, the experiments on the cooling-cffect c, when combined with those of the specific heat s, showed that the product sc was a func- tion of the temperature only, and gave the simple expression for the total heat, h=s,t —scp+b (19) for dry steam at any pressure p. the values for dry saturated steam, given by putting the saturation pressure p in this expression, while differing materially from regnault’s formula, gave good agreement with the experiments (see 27.902) of dicterici at o° c., and of grif- fiths at 30° and 40°c., when the constant b was deduced from joly’s observations at 100° c, with the aid of the experiments on the specific heat of water. this formula was closely confirmed by the observa- tions of hi. henning (ann. phys., 21, p. 849, 1906) on the latent heat between 30° and roo°c. ilis later observations (ann. phys., 29, p. 441, 1909) also gave good agreement with the same curve at 180° c., but showed a discontinuity at 120°c., which may be attrib- uted to inevitable experimental errors in such difficult work. at higher temperatures, up to 260° c., equation (19) received theoreti- cal confirmation from the formula for the latent heat proposed by m. thiesen, namely, l=l,(¢,—e)'*, based on the vanishing of the latent heat at the critical temperature f,. as first applied by thiesen himself (ann. phys., 9, p. 80, 1902) to the case of steam, with 365°c. for the critical temperature, this formula gave results which were much too low for the latent heat. it was shown, how- ever, by traube and teichner (ann. phys., 13, p. 620, 1904) that the true value of ¢, was 374°c., which brought thiesen’s formula into agreement with (19) to less than i in 1,000 all the way from o° to 260°c,, when the constants were properly determined from the known values at 0°, 100°, 180° and 374°, giving the result log l =1-9638+0-3151 log (374—#) (20) in the logarithmic form as required for practical calculations. the importance of this formula arises from the fact that direct deter- minations of h, (for dry saturated steam) become exceedingly difficult and uncertain at temperatures above 180°c., owing to errors from leakage and wetness, and that a formula of this type has been verified for many other substances in the critical region, so that it affords the best guide to the probable variation of h, between 200° and 374°c. the throttling experiments showed that there must be a consid- erable variation of s with pressure, corresponding to the variation of sc with temperature. but the experiments on the adiabatic expansion of dry steam showed that the index »+-1 in the equation p/t*+l=constant, was very nearly constant and equal to 13/3 over a wide range of p and t. since so/r =13/3, it followed that the total heat of dry steam must be expressible in the form (21) h =(13@/3)p(v—8)+a8p+b giving the convenient expression for the volume of dry steam, v = (3/13@) (ei —b)/p+ 106/13 (22) it also followed that the coaggregation volume c=co(to/t)" in the equation v—b=rt/ep—c (23) must vary with temperature according to the index »=10/3, giving for the variation of s and c, in terms of c, the formulae sc =a(n+1)e—ab (24) s=sotan(n+r)cp/t (25) it was obvious that these could not apply accurately at high pres- sures in the critical region, but they afford ample accuracy for all purposes in the pressures required in steam-engine practice. the munich experiments (forsch. ver, dent. ing., 21, 1905) by o. knoblauch, r. linde and kk. klebe, on the volume of steam, proved to be quite inconsistent with the well-known equation of zeuner, then commonly accepted, but showed the most remarkable agreement up to 180°c. with formula (23) deduced from the throt- tling experiments. the variation of s with pressure given by (25), as predicted by the experiments on c, was qualitatively confirmed by the experi- ments of o. knoblauch and m. jakob (forsch. ver. deut. ing., 36, p. 109, 1906) extending to 8 atmospheres. heat knoblauch and his collaborators have since extended their observations on the specific heat up to 39 atm. (425 |lb.), and have published steam tables (olden- bourg, munich, 1923) extrapolated to 60 atm. (850 lb.). these results are in very fair agreement with (19), (22) and (23), especially near the limit of 850 ib. at a moderate degree of superheat, but show a rapid fall of total heat on approaching saturation. (see also **world-power,”’ for may and june 1924, and june 1925.) callendar has since extended his observations with the dilferential throttling calo- rimeter to 1,000 1b. pressure, and finds (as explained in properties of steam, p. 83) that the assumption that the cooling-cffect c is a function of the temperature only gives better agreement with experiment at high pressures near saturation than the joule-thomson equation (23), though both are in practical aceord at moderate pressures or high superheats. there is always a difficulty in securing accurate observations of the total heat or specific heat of safuraied steam owing to syste- matic errors of wetness, but results accurate to i in 1,000 may readily be obtained at a moderate degree of superheat. adiabatic heat-drap.—the change of total heat h in frictionless adiabatic expansion or compression is frequently of considerable interest as representing the work done by or on the fluid in the ideal case, when there ts no internal friction, and when no heat is supplied or lost externally. if the laws of thermodynamics are summarised in the form dq =tdb =dii —avdp (26) in which dq represents heat supplied per unit mass by friction or otherwise, we observe that, in the case of isentropic flow, for which db =o, the change of hl is equal to the integral of avdp along the adiabatic, which is readily obtained by substituting for v in terms of hand p from (22) or (17), for any given initial state and final pres- sure. we may also obtain the general expression for & from those for ei and v by integrating d@’=dl1/t—(av/t)dp. these expres- sions may be put in various forms according to the purpose for which they are required. one of the most useful for dry steam is dig = (h’ —h’)¢ =(h’ —b—abp’)(1—-t”/t’) +ab(p’-—p”) (27) in which il’, p’, ‘i’, and h”, p’, t’, represent the initial and final states. an exact expression for the adiabatic heat drop dh¢, in the case of wet saturated steam, is readily obtained in terms of h’ and t’, t’’. but in practice it is usually more convenient to tabu- late h} and #, and the gibbs’ function g=t@—hi, which has the advantage of being a simple function of the temperature only, and independent of the wetness for a mixture of water and steam in any proportions. from the definition of g, if ¢ is constant at its initial value ¢”, we obtain immediately the convenient expressions dh¢ = (t’ —t")&’ —g'+g” = h’-—h.” + t" (@,” — 9’) (28) the first expression is general, and is readily applied if g’ and g” are tabulated. the second is obtained by substituting for g’ and g” in terms of h and ¢, but is applicable only if the final state is saturated, so that h.’’ and ¢,” are the tabulated values for dry saturated steam. effects of supersaturation.—for the general theory of the behav- iour of a vapour when cooled below the saturation tempcrature without condensation see 27.898-9. the state of supersaturation is very common, in rapid expansion, and has proved to be of some practical importance, as affecting the discharge through a nozzle, and the efficiency of a turbine. it appears that steam usually fol- lows the dry adiabatic, p/t's =constant, for some distance below the saturation point. the drop of temperature is about three times as rapid as along the wet adiabatic, and the volume is smaller than that of saturated steam at the same p and h. the heat-drop, and the velocity generated, are also smaller, for a given pressure drop, than in the case of steam which is assumed to remain in the equilib- rium state of saturation throughout the expansion. if the initial steam is dry saturated, it usually remains dry for some distance beyond the throat of a nozzle, so that the discharge, as given by equation (12), is obtained from the dry adiabatic, by substituting (dp/dv)¢ =1-3p/v at the throat, which leads to values about 5% larger than those given by the equations for wet steam. this is confirmed by experiment, and is represented by the numerical formula for the discharge m/x, in lb. per sec. per sq. in. of throat, when p’ is in ib./sq. in. and v’ in cu. ft./ib. in the initial state, m/xe=0-3155(p7v)#, pa/p’ = 0-545 (29) in which the small quantity 6 is neglected as being usually beyond the limits of possible accuracy of measurement. the defect of heat-drop on reaching the throat is about 5°. hf the steam continued to follow the dry adiabatic to low pressures, the defect of heat-drop would often reach 20%, which would be very serious. but soon after passing the throat, the coaggregated mole- cules begin to act as condensation nuclei, according to kelvin's equation (see 27.898). when this limit is reached, the condensation takes the form of a very thick fog of exceedingly fine particles, and is extremely rapid, owing to the enormous number of nuclei avail- able, about ro” per ib. of steam. hf the expansion is relatively slow, the steam is transformed into the saturated state, and remains nearly saturated for the rest of the expansion. but if the expansion ts very rapid, asin an expanding nozzle at a velocity of 3 or 4,000 ft./sec. the steam will remain near the supersaturation limit with a loss of heatdrop amounting to nearly 8° at low pressures, involving a cor- responding loss of efficiency. according to wilson's experiments at low pressures (see 27.899), the supersaturation limit is reached when the pressure is about 8 times the normal saturation pressure corresponding to the actual temperature of the steam. the equiva- lent wetness of the steam at this point, when transformed to the heating and saturated state at the same p and h, would be about 3%. this appears to be confirmed by turbine tests at these pressures, but wu- son's experiments do not afford any direct evidence with regard to the limit at which condensation starts at higher pressures. it appears on theoretical grounds that the pressure ratio corresponding to the supersaturation limit should not be so high as 8 at hick pressures, which would require an excessive increase in the drop of temperature and in the equivalent wetness of the steam at high pressures. there is some evidence that the equivalent wetness at the super- saturation limit is the same, namely 3‘%, at high as at low pres- sures. this would permit a very simple method of calculation, but more experimental tests are required to decide the point. the effect of initial superheat in improving the efficiency of a turbine cannot be satisfactorily explained on the older theory that the steam is in the equilibrium state of saturation throughout the expansion, but is a necessary consequence of the phenomenon of supersaturation. the loss due to supersaturation may be entirely eliminated if the supcrheat is sufficient to prevent supersaturation. in any case the loss will be greatly reduced by superheat, and the results of calcu- lation appear to indicate that the improvement of efficiency may be exactly accounted for in this way. this point has been very fully discussed by h. m. martin, in “ a new theory of the steam turbine ” (/ingineering, vol. 106, 1918); and also by h. l. callendar, properties of steam, pp. 305-12. references.—on the practical side, sir j, a. ewing’s mechanical production of cold and thermodynantics for engineers (1920); on the theoretical side, h. s. carslaw, fourier’s series and integrals; and j. h. jeans, dynamical theory of gases. for experimental details it is always necessary to refer to the original papers, but physical and chemical constants by g. w.c. kaye and t. h. laby (1921) gives a handy and up-to-date summary of numerical results. chiut. ca)