Q'T'

q't', and therefore serve to measure the area swept over like the wheel in the machine already described. the turning of the rod will also produce slipping of the wheel, but it will be seen without difficulty that this will cancel during a cyclical motion of the rod, provided the rod does not perform a whole rotation. [illustration: fig. 15.] the first planimeter was made on the following principles:--a frame ff (fig. 15) can move parallel to ox. it carries a rod tt [sidenote: early forms.] movable along its own length, hence the tracer t can be guided along any curve atb. when the rod has been pushed back to q'q, the tracer moves along the axis ox. on the frame a cone vcc' is mounted with its axis sloping so that its top edge is horizontal and parallel to tt', whilst its vertex v is opposite q'. as the frame moves it turns the cone. a wheel w is mounted on the rod at t', or on an axis parallel to and rigidly connected with it. this wheel rests on the top edge of the cone. if now the tracer t, when pulled out through a distance y above q, be moved parallel to ox through a distance dx, the frame moves through an equal distance, and the cone turns through an angle d[theta] proportional to dx. the wheel w rolls on the cone to an amount again proportional to dx, and also proportional to y, its distance from v. hence the roll of the wheel is proportional to the area ydx described by the rod qt. as t is moved from a to b along the curve the roll of the wheel will therefore be proportional to the area aa'b'b. if the curve is closed, and the tracer moved round it, the roll will measure the area independent of the position of the axis ox, as will be seen by drawing a figure. the cone may with advantage be replaced by a horizontal disk, with its centre at v; this allows of y being negative. it may be noticed at once that the roll of the wheel gives at every moment the area a'atq. it will therefore allow of registering a set of values of [integral,a:x] ydx for any values of x, and thus of tabulating the values of any indefinite integral. in this it differs from amsler's planimeter. planimeters of this type were first invented in 1814 by the bavarian engineer hermann, who, however, published nothing. they were reinvented by prof. tito gonnella of florence in 1824, and by the swiss engineer oppikofer, and improved by ernst in paris, the astronomer hansen in gotha, and others (see henrici, _british association report_, 1894). but all were driven out of the field by amsler's simpler planimeter. [illustration: fig. 16.] [illustration: fig. 17.] altogether different from the planimeters described is the hatchet planimeter, invented by captain prytz, a dane, and made by herr [sidenote: hatchet planimeters.] cornelius knudson in copenhagen. it consists of a single rigid piece like fig. 16. the one end t is the tracer, the other q has a sharp hatchet-like edge. if this is placed with qt on the paper and t is moved along any curve, q will follow, describing a "curve of pursuit." in consequence of the sharp edge, q can only move in the direction of qt, but the whole can turn about q. any small step forward can therefore be considered as made up of a motion along qt, together with a turning about q. the latter motion alone generates an area. if therefore a line oa = qt is turning about a fixed point o, always keeping parallel to qt, it will sweep over an area equal to that generated by the more general motion of qt. let now (fig. 17) qt be placed on oa, and t be guided round the closed curve in the sense of the arrow. q will describe a curve osb. it may be made visible by putting a piece of "copying paper" under the hatchet. when t has returned to a the hatchet has the position ba. a line turning from oa about o kept parallel to qt will describe the circular sector oac, which is equal in magnitude and sense to aob. this therefore measures the area generated by the motion of qt. to make this motion cyclical, suppose the hatchet turned about a till q comes from b to o. hereby the sector aob is again described, and again in the positive sense, if it is remembered that it turns about the tracer t fixed at a. the whole area now generated is therefore twice the area of this sector, or equal to oa. ob, where ob is measured along the arc. according to the theorem given above, this area also equals the area of the given curve less the area osbo. to make this area disappear, a slight modification of the motion of qt is required. let the tracer t be moved, both from the first position oa and the last ba of the rod, along some straight line ax. q describes curves of and bh respectively. now begin the motion with t at some point r on ax, and move it along this line to a, round the curve and back to r. q will describe the curve dosbed, if the motion is again made cyclical by turning qt with t fixed at a. if r is properly selected, the path of q will cut itself, and parts of the area will be positive, parts negative, as marked in the figure, and may therefore be made to vanish. when this is done the area of the curve will equal twice the area of the sector rde. it is therefore equal to the arc de multiplied by the length qt; if the latter equals 10 in., then 10 times the number of inches contained in the arc de gives the number of square inches contained within the given figure. if the area is not too large, the arc de may be replaced by the straight line de. to use this simple instrument as a planimeter requires the possibility of selecting the point r. the geometrical theory here given has so far failed to give any rule. in fact, every line through any point in the curve contains such a point. the analytical theory of the inventor, which is very similar to that given by f.w. hill (_phil. mag._ 1894), is too complicated to repeat here. the integrals expressing the area generated by qt have to be expanded in a series. by retaining only the most important terms a result is obtained which comes to this, that if the mass-centre of the area be taken as r, then a may be any point on the curve. this is only approximate. captain prytz gives the following instructions:--take a point r as near as you can guess to the mass-centre, put the tracer t on it, the knife-edge q outside; make a mark on the paper by pressing the knife-edge into it; guide the tracer from r along a straight line to a point a on the boundary, round the boundary, [v.04 p.0978] and back from a to r; lastly, make again a mark with the knife-edge, and measure the distance c between the marks; then the area is nearly cl, where l = qt. a nearer approximation is obtained by repeating the operation after turning qt through 180° from the original position, and using the mean of the two values of c thus obtained. the greatest dimension of the area should not exceed 1⁄2l, otherwise the area must be divided into parts which are determined separately. this condition being fulfilled, the instrument gives very satisfactory results, especially if the figures to be measured, as in the case of indicator diagrams, are much of the same shape, for in this case the operator soon learns where to put the point r. integrators serve to evaluate a definite integral [integral,a:b] f(x)dx. if we plot out [sidenote: integrators.] the curve whose equation is y = f(x), the integral [integral]ydx between the proper limits represents the area of a figure bounded by the curve, the axis of x, and the ordinates at x=a, x=b. hence if the curve is drawn, any planimeter may be used for finding the value of the integral. in this sense planimeters are integrators. in fact, a planimeter may often be used with advantage to solve problems more complicated than the determination of a mere area, by converting the one problem graphically into the other. we give an example:-- [illustration: fig. 18.] let the problem be to determine for the figure abg (fig. 18), not only the area, but also the first and second moment with regard to the axis xx. at a distance a draw a line, c'd', parallel to xx. in the figure draw a number of lines parallel to ab. let cd be one of them. draw c and d vertically upwards to c'd', join these points to some point o in xx, and mark the points c_1d_1 where oc' and od' cut cd. do this for a sufficient number of lines, and join the points c_1d_1 thus obtained. this gives a new curve, which may be called the first derived curve. by the same process get a new curve from this, the second derived curve. by aid of a planimeter determine the areas p, p_1, p_2, of these three curves. then, if [=x] is the distance of the mass-centre of the given area from xx; [=x]_1 the same quantity for the first derived figure, and i = ak2 the moment of inertia of the first figure, k its radius of gyration, with regard to xx as axis, the following relations are easily proved:-- p[=x] = ap_1; p_1[=x]_1 = ap_2; i = ap_1[=x]_1 = a2p_1p_2; k2 = [=x][=x]_1, which determine p, [=x] and i or k. amsler has constructed an integrator which serves to determine these quantities by guiding a tracer once round the boundary of the given figure (see below). again, it may be required to find the value of an integral [integral]y[phi](x)dx between given limits where [phi](x) is a simple function like sin nx, and where y is given as the ordinate of a curve. the harmonic analysers described below are examples of instruments for evaluating such integrals. [illustration: fig. 19.] [illustration: fig. 20.] amsler has modified his planimeter in such a manner that instead of the area it gives the first or second moment of a figure about an axis in its plane. an instrument giving all three quantities simultaneously is known as amsler's integrator or moment-planimeter. it has one tracer, but three recording wheels. it is mounted on a [sidenote: amsler's integrator.] carriage which runs on a straight rail (fig. 19). this carries a horizontal disk a, movable about a vertical axis q. slightly more than half the circumference is circular with radius 2a, the other part with radius 3a. against these gear two disks, b and c, with radii a; their axes are fixed in the carriage. from the disk a extends to the left a rod ot of length l, on which a recording wheel w is mounted. the disks b and c have also recording wheels, w_1 and w_2, the axis of w_1 being perpendicular, that of w_2 parallel to ot. if now t is guided round a figure f, o will move to and fro in a straight line. this part is therefore a simple planimeter, in which the one end of the arm moves in a straight line instead of in a circular arc. consequently, the "roll" of w will record the area of the figure. imagine now that the disks b and c also receive arms of length l from the centres of the disks to points t_1 and t_2, and in the direction of the axes of the wheels. then these arms with their wheels will again be planimeters. as t is guided round the given figure f, these points t_1 and t_2 will describe closed curves, f_1 and f_2, and the "rolls" of w_1 and w_2 will give their areas a_1 and a_2. let xx (fig. 20) denote the line, parallel to the rail, on which o moves; then when t lies on this line, the arm bt_1 is perpendicular to xx, and ct_2 parallel to it. if ot is turned through an angle [theta], clockwise, bt_1 will turn counter-clockwise through an angle 2[theta], and ct_2 through an angle 3[theta], also counter-clockwise. if in this position t is moved through a distance x parallel to the axis xx, the points t_1 and t_2 will move parallel to it through an equal distance. if now the first arm is turned through a small angle d[theta], moved back through a distance x, and lastly turned back through the angle d[theta], the tracer t will have described the boundary of a small strip of area. we divide the given figure into [v.04 p.0979] such strips. then to every such strip will correspond a strip of equal length x of the figures described by t_1 and t_2. the distances of the points, t, t_1, t_2, from the axis xx may be called y, y_1, y_2. they have the values y = l sin [theta], y_1 = l cos 2[theta], y_2 = -l sin 3[theta], from which dy = l cos [theta].d[theta], dy_1 = - 2l sin 2[theta].d[theta], dy_2 = - 3l cos 3[theta].d[theta]. the areas of the three strips are respectively da = xdy, da_1 = xdy_1, da_2 = xdy_2. now dy_1 can be written dy_1 = - 4l sin [theta] cos [theta]d[theta] = - 4 sin [theta]dy; therefore da_1 = - 4 sin [theta].da = - (4/l) yda; whence a_1 = - 4/l [integral]yda = - 4/l a[=y], where a is the area of the given figure, and [=y] the distance of its mass-centre from the axis xx. but a_1 is the area of the second figure f_1, which is proportional to the reading of w_1. hence we may say a[=y] = c_1w_1, where c_1 is a constant depending on the dimensions of the instrument. the negative sign in the expression for a_1 is got rid of by numbering the wheel w_1 the other way round. again dy_2 = - 3l cos [theta] {4 cos2 [theta] - 3} d[theta] = - 3 {4 cos2 [theta] - 3} dy = - 3 {(4/l2) y2 - 3} dy, which gives da_2 = - (12/l2)y2da + 9da, and a_2 = - (12/l2) [integral]y2da + 9a. but the integral gives the moment of inertia i of the area a about the axis xx. as a_2 is proportional to the roll of w_2, a to that of w, we can write i = cw - c_2 w_2, a[=y] = c_1 w_1, a = c_c w. if a line be drawn parallel to the axis xx at the distance [=y], it will pass through the mass-centre of the given figure. if this represents the section of a beam subject to bending, this line gives for a proper choice of xx the neutral fibre. the moment of inertia for it will be i + a[=y]2. thus the instrument gives at once all those quantities which are required for calculating the strength of the beam under bending. one chief use of this integrator is for the calculation of the displacement and stability of a ship from the drawings of a number of sections. it will be noticed that the length of the figure in the direction of xx is only limited by the length of the rail. this integrator is also made in a simplified form without the wheel w_2. it then gives the area and first moment of any figure. while an integrator determines the value of a definite integral, hence a [sidenote: integraphs.] mere constant, an integraph gives the value of an indefinite integral, which is a function of x. analytically if y is a given function f(x) of x and y = [integral,c:x]ydx or y = [integral]ydx + const. the function y has to be determined from the condition dy/dx = y. graphically y = f(x) is either given by a curve, or the graph of the equation is drawn: y, therefore, and similarly y, is a length. but dy/dx is in this case a mere number, and cannot equal a length y. hence we introduce an arbitrary constant length a, the unit to which the integraph draws the curve, and write dy/dx = y/a and ay = [integral]ydx. now for the y-curve dy/dx = tan [phi], where [phi] is the angle between the tangent to the curve, and the axis of x. our condition therefore becomes tan [phi] = y / a. [illustration: fig. 21.] this [phi] is easily constructed for any given point on the y-curve:--from the foot b' (fig. 21) of the ordinate y = b'b set off, as in the figure, b'd = a, then angle bdb' = [phi]. let now db' with a perpendicular b'b move along the axis of x, whilst b follows the y-curve, then a pen p on b'b will describe the y-curve provided it moves at every moment in a direction parallel to bd. the object of the integraph is to draw this new curve when the tracer of the instrument is guided along the y-curve. the first to describe such instruments was abdank-abakanowicz, who in 1889 published a book in which a variety of mechanisms to obtain the object in question are described. some years later g. coradi, in zurich, carried out his ideas. before this was done, c.v. boys, without knowing of abdank-abakanowicz's work, actually made an integraph which was exhibited at the physical society in 1881. both make use of a sharp edge wheel. such a wheel will not slip sideways; it will roll forwards along the line in which its plane intersects the plane of the paper, and while rolling will be able to turn gradually about its point of contact. if then the angle between its direction of rolling and the x-axis be always equal to [phi], the wheel will roll along the y-curve required. the axis of x is fixed only in direction; shifting it parallel to itself adds a constant to y, and this gives the arbitrary constant of integration. in fact, if y shall vanish for x = c, or if y = [integral,c:x]ydx, then the axis of x has to be drawn through that point on the y-curve which corresponds to x = c. [illustration: fig. 22.] in coradi's integraph a rectangular frame f_1f_2f_3f_4 (fig. 22) rests with four rollers r on the drawing board, and can roll freely in the direction