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AQB

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Originally appearing in Volume V04, Page 981 of the 1911 Encyclopedia Britannica.
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AQB through which the rod has turned. The wheel will roll over an arc c9, where c is the distance of the wheel from Q. The "roll" is now w = c9 ; hence the area generated is P II •'-, w, and is again determined by w. B Next let the rod be moved FIG. 7. parallel to itself, but in a direction not perpendicular to itself (fig. 8). The wheel will now not simply roll. Consider a small motion of the rod from QT to Q'T'. This may be resolved into the motion to RR' perpendicular to the rod, whereby the rectangle QTR'R is generated, and the sliding of the rod along itself from RR' to Q'T'. During this second step no area will be generated. During the first step the roll of the wheel will be QR, whilst during the second step there will be no roll at all. The roll of the wheel will there-fore measure the area of the rectangle which equals the parallelogram QTT'Q'. If the whole motion of the rod be considered as made up of a very great number of small steps, each resolved as stated, it will be seen that the roll again measures the area generated. But it has to be noticed that now the wheel does not only roll, but also slips, over the paper. This, as will be pointed out later, may introduce an error in the reading. We can now investigate the most general motion of the rod. We again resolve the motion into a number of small steps. Let (fig. 9) AB be one position, CD the next after a step so small that the arcs AC and BD over which the ends have passed may be considered as straight lines. The area generated is ABDC. This motion we resolve into a step from AB to CB', parallel to AB and a turning about C from CB' to CD, steps such as have been investigated. During the first, the "roll" will be p the altitude of the parallelogram; during the second will be CO. Therefore w=p+c9. The area generated is 1p+-129, or, expressing p in -terms of w, 1w+(4I -lc)9. For a finite motion we get the area equal to the sum of the areas generated during the different steps. But the wheel will continue rolling, and give the whole roll as the sum of the rolls for the successive steps. Let then w denote the whole roll (in 4 fig. --~8 so), and let a denote the sum of all the small turnings 0; then the area is P=lw+(-IP-lc)a. (I) Here a is the angle which the A ( B last position of the rod makes FIG. 9. with the first. In all applica- tions of the planimeter the rod is brought back to its original position. Then the angle a is either zero, or it is 2r if the rod has been once turned quite round. Hence in the first case we have P=lw . (2a) and w gives the area as in case of a rectangle. In the other case P=lw+lC . (2b) where C= (1l-c)2ar, if the rod has once turned round. The number C will be seen to be always the same, as it depends only on the dimen- A sions of the instrument. Hence now again the area is FIG. so. determined bywif C is known. Thus it is seen that the area generated by the motion of the rod can be measured by the roll of the wheel ; it remains to show how any given area can be generated by the rod. Let the rod move in any manner but return to its original position. Q and T then describe closed curves. Such motion may be called cyclical. Here the theorem holds If a rod QT performs a cyclical motion, then the area generated equals the difference of the areas enclosed by the paths of T and Q respectively. The truth of this proposition will be seen from a figure. In fig. II FIG II. different posit;ons of the moving rod QT have been marked, and its motion can be easily followed. It will be seen that every part of the area TT'BB' will be passed over once and always by a forward motion of the rod, whereby the wheel will increase its roll. The area AA'QQ' will also be swept over once, but with a backward roll; it must therefore be counted as negative. The area between the curves is passed over twice, once with a forward and once with a backward roll; it therefore counts once positive and once negative; hence not at all. In more complicated figures it may happen that the area within one of the curves, say TT'BB', is passed over several times, but then it will be passed over once more in the forward direction than in the backward one, and thus the theorem will still hold. To use Amsler's planimeter, place the pole 0 on the paper outside the figure to be measured. Then the area generated by QT is that of the figure, because the point Q moves on an arc of a circle to and fro enclosing no area. At the same time the rod comes back without making a complete rotation. We have therefore in formula (I), a =o' and hence P=1w, T w 0 8 FIG. 6. D which is read off. But if the area is too large the pole 0 may be placed within the area. The rod describes the area between the boundary of the figure and the circle with radius r=OQ, whilst the rod turns once completely round, making a =27r. The area measured by the wheel is by formula (t), lw+ (Zl2-lc)21r. To this the area of the circle 7rr2 must be added, so that now P=lw+('-2l2-lc)2a+1rr2, or P =1w+C, where C=(Zl2-lc)2ir+1rr2 is a constant, as it depends on the dimensions of the instrument alone. This constant is given with each instrument. Amsler's planimeters are made either with a rod QT of fixed length, which gives the area therefore in terms of a fixed unit, say in square inches, or else the rod can be moved in a sleeve to which the arm OQ is hinged (fig. 13). This makes it possible to change the unit lu, which is proportional to 1. In the planimeters described the recording or integrating apparatus is a smooth wheel rolling on the paper or on some other surface. Amsler has described another recorder, viz. a wheel with a sharp edge. This will roll on the paper but not slip. Let the rod QT carry with it an arm CD perpendicular to it. Let there be mounted on it a wheel W, which can slip along and turn about it. If now QT is moved parallel to itself to Q'T', then W will roll without slipping parallel to QT, and slip along CD. This amount of slipping will equal the perpendicular distance between QT and 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. The first planimeter was made on the following principles :—A frame FF (fig. 15) can move parallel to OX. It carries a rod TT Early movable along its torros. 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 5. dx. The wheel W rolls on the cone to an amount again proportional to dx, and also proportional to iits distance from V. Hence the roll of the wheel is proportional to he 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 roundit, 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 ATQ. It will therefore allow of registering a set of values off zydx 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. Altogether different from the planimeters described is the hatchet planimeter, invented by Captain Prytz, a Dane, and made by Herr Cornelius Knudson in Copenhagen. It Hatchet consists of a single meters. 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 FIG. 16. curve, Q will follow, describ- ing 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 0, 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 0 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 c __ 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 there-fore 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 to times the number of inche 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, 13. and back from A to R; lastly, make again a mark with the knife- carriage which runs on a straight rail (fig. 19). This carries a horiedge, and measure the distance c between the marks; then the zontal disk A, movable about a vertical axis Q. Slightly more than area is nearly cl, where 1= QT. A nearer approximation is obtained half the circumference is circular with radius 2a, the other part with by repeating the operation after turning QT through 18o° 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 11, 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 cf the same shape, for in this case the operator soon learns where to put the poirt R. Integrators serve to evaluate a de- b finite f.f(x)dx. If we plot out Irate- the curve whose equation is grators. y= f(x), the integral f 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: 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 0 in XX, and mark the points C1Di where OC' and OD' cut Cll. Do this for a sufficient number of lines, and join the points C1Di 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, Pi, P2, of these three curves. Then, if is the distance of the mass-centre of the given area from XX; xi 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: Pi = aPi ; Pi–xi =aP2; I=aPixi=a2PiP2; k2=xx1, which determine P, x and I or k. Amsler has constructed an integrator which serves to determine these quantities by guiding a C' D n n` G e j\, i 1 0 tracer once round the boundary of the given figure (see below). A ain, it may be required to find the value of an integral J y/(x)dx between given limits where ¢(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. 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 radius 3a. Against these gear two disks, B and C, with radii a; their axes are fixed in the carriage. From the disk A ex- Amsler's tends to the left a rod OT of length 1, on which a record- Inteing wheel W is mounted. The disks B and C have also grator. recording wheels, Wi and W2, the axis of Wi being perpendicular, that of W2 parallel to OT. If now T is guided round a figure F, 0 will move to and fro in a straight line. This part is there-fore 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 Ti and T2, 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 Ti and T2 will describe closed curves, Fi and F2, and the " rolls " of Wi and W2 will give their areas Ai and A2. Let XX (fig. 20) denote the line, parallel to the rail, on which 0 moves; then when T lies on this line, the arm BT1 is perpendicular to XX, and CT2 parallel to it. If OT is turned through an angle 0, clockwise, BTi will turn counter-clockwise through an angle 20, and CT2 through an angle 30, also counter-clockwise. If in this position T is moved through a distance x parallel to the axis XX, the points Ti and T2 will move parallel to it through an equal distance. If now the first arm is turned through a small angle do, moved back through a distance x, and lastly turned back through the angle do, the tracer 'I will have described the boundary of a small strip of area. We divide the given figure into such strips. Then to every such strip will correspond a strip of equal length x of the figures described by T1 and T2. The distances of the points, T, T2, from the axis XX may be called y, yi, y2. They have the values y=lsin0,yi=lcos20,y2=—1 sin 30, from which dy =1 cos0.de, dyi=—2l sin20.de, dy2=—3lcos30.de. The areas of the three strips are respectively dA=xdy, dA,=xdyi, dA2=xdyi. Now dyl can be written dyi = -41 sin 0 cos ede = -4 sin edy; therefore dA, _ -4 sin e.dA= — ZydA; A,= —L fydA—lAy, where A is the area of the given figure, and y the distance of its mass-centre from the axis XX. But A, is the a ea of the second figure F,, which is proportional to the reading of W1. Hence we may say Ay = Ciwi, where C, is a constant depending on the dimensions of the instrument. The negative sign in the expression for Al is got rid of by numbering the wheel W, the other way round. Again dye = -3l cos 0 {4 cos' 0—3d0—3{4 cos' 0 -31dy =—3)y2—3 dy, dA2= — l2 y2dA+9dA, whence and A2 = — l2 f y2dA+9A. But the integral gives the moment of inertia I of the area A about the axis XX. As A2 is proportional to the roll of w2, A to that of W, we can write I = Cw — C2w2, Ay = Cea,, A = Ccw. 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 he I+Ay. 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 WW/2. It then gives the area and first moment of any figure. While an integrator determines the value of a definite integral, hence a late- mere constant, an integraph graphs. Fives 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 =r ydx or Y = f ydx+const. Fl the function Y has to be determined from the condition dY ax=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 dti 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 dx = ¢ and aY =f ydx. Now for the Y-curve d = tan ¢, where is the angle between the tangent to the curve, and the axis of x. Our condition therefore becomes This is easily constructedtanfor0a= nyagi•ven 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'=¢. 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 qaestion 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 ¢, 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 = xydx, then the axis of x has to be drawn through that point on the y-curve which corresponds to x=c. In Coradi's integraph a rectangular frame_F1F2F3F4 (fig. 22) rests with four rollers R on the drawing board, and can roll freely in the direction OX, which will be called the axis of the instrument. On the front edge F1F2 travels a carriage AA' supported at A' on another rail. A bar DB can turn about D, fixed to the frame in its axis, and slide through a point B fixed in the carriage AA'. Along it a block K can slide. On the back edge F3F4 of the frame another carriage C travels. It holds a vertical spindle with the knife-edge wheel at the bottom. At right angles to the plane of the wheel, the spindle has an arm GH, which is kept parallel to a which gives x x a g' D P sr.. r'7C sq.. or ma. 0_` R - - A C R Ft similar arm attached to K perpendicular to DB. The plane of the knife-edge wheel r is therefore always parallel to DB. If now the point B is made to follow a curve whose y is measured from OX, we have in the triangle BDB', with the angle at D, tan =y/a, ct) where a= DB' is the constant base to which the instrument works. The point of contact of the wheel r or any point of the carriage C will therefore always move in a direction making an angle q5 with the axis of x, whilst it moves in the x-direction through the same distance as the point B on the y-curve—that is to say, it will trace out the integral curve required, and so will any point rigidly connected with the carriage C. A pen P attached to this carriage will therefore draw the integral curve. Instead of moving B along the y-curve, a tracer T fixed to the carriage A is guided along it. For using the instrument the carriage is placed on the drawing-board with the front edge parallel to the axis of y, the carriage A being clamped in the central position with A at E and B at B' on the axis of x. The tracer is then placed on the x-axis of the y-curve and clamped to the carriage, and the instrument is ready for use. As it is convenient, to have the integral curve placed directly opposite to the y-curve so that corresponding values of y or Y are drawn on the same line, a pen P' is fixed to C in a line with the tracer. Boys' integraph was invented during a sleepless night, and during the following days carried out as a working model, which gives highly satisfactory results. It is ingenious in its simplicity, and a direct realization as a mechanism of the principles explained in connexion with fig. 21. The line B'B is represented by the edge of an ordinary T-square sliding against the edge of a drawing-board. The points B and P are connected by two rods BE and EP, jointed at E. At B, E and P are small pulleys of equal diameters. Over these an endless string runs, ensuring that the pulleys at B and P always turn through equal angles. The pulley at B is fixed to a rod which passes through the point D, which itself is fixed in the T-square. The pulley at P carries the knife-edge wheel. If then B and P are kept on the edge of the T-square, and B is guided along the curve, the wheel at P will roll along the Y-curve, it having been originally set parallel to BD. To give the wheel at P sufficient grip on the paper, a small loaded three-wheeled carriage, the knife-edge wheel P being one of its wheels, is added. If a piece of copying paper is inserted between the wheel P and the drawing paper the Y-curve is drawn very sharply. Integraphs have also been constructed, by aid of which ordinary differential equations, especially linear ones, can be solved, the solution being given as a curve. The first suggestion in this direction was made by Lord Kelvin. So far no really useful instrument has been made, although the ideas seem sufficiently developed to enable a skilful instrument-maker to produce one should there be sufficient demand for it. Sometimes a combination of graphical work with an integraph will serve the purpose. This is the case if the variables are separated, hence if the equation Xdx+Ydy=O has to be integrated where X=p(x), Y=0(y) are given as curves. If we write au =f xdx, av = f Ydy, then u as a function of x, and v as a function of y canbe graphically found by the integraph. The general solution is then u+v=c with the condition, for the determination for c, that y=yo, for x=xo. This determines c=uo+vo, where uo and it, are known from the graphs of u and v. From this the solution as a curve giving y a function of x can be drawn:—For any x take u from its graph, and find the y for which v=c—u, plotting these y against their x gives the curve required. If a periodic function y of x is given by its graph for one period c, it can, according to the theory of Fourier's Series; be Harmonic expanded in a series. analysers. y =Ad+Altos a+A2cos2 B+ ... +A„cos ne+ .. . +Basin B +Basin2 8+ . . . +Basin nO+ . . . 27X where B = c The absolute term Ao equals the mean ordinate of the curve, and can therefore be determined by any planimeter. The other coefficients are A= 1 y cos no.dO; =1 J y sin no.dO. 7rJ o r 0 A harmonic analyser is an instrument which determines these integrals, and is therefore an integrator. The first instrument of this kind is due to Lord Kelvin (Prot. Roy. Soc., vol. xxiv., 1876). Since then several others have been invented (see Dyck's Catalogue; Henrici, Phil. Mdg., July 1894; Phys. Soc., 9th March; Sharp, Phil. Mag., July 1894; Phys. Soc., 13th April). In Lord Kelvin's instrument the curve to be analysed is drawn on a cylinder whose circumference equals the period c, and the sine and cosine terms of the integral are introduced by aid of simple harmonic motion. Sommerfeld and Wiechert, of Konigsberg, avoid this motion by turning the cylinder about an axis perpendicular to that of the cylinder. Both these machines are large, and practically fixtures in the room where they are used. The first has done good work in the Meteorological Office in London in the analysis of meteorological curves. Quite different and simpler constructions can be used, if the integrals determining An and B,, be integrated by parts. This gives nA,, = — J sin nO.dy; nB,, = cos nO.dy. o An analyser presently to be described, based on these forms, has been constructed by Coradi in Zurich (1894). Lastly, a most powerful analyser has been invented by Michelson and Stratton (U.S.A.) (Phil. Meg., 1898), which will also be described. The Henrici-Coradi analyser has to add up the values of dy. sin nO and dy. cos nO. But these are the components of dy in two directions perpendicular to each other, of which one makes an angle nO with the axis of x or of B. This decomposition can he performed by Amsler's registering wheels. Let two of these be mounted, perpendicular to each other, in one horizontal frame which can be Ifs- turned about a vertical axis, the wheels resting on the paper on which the curve is drawn. When the tracer is placed on the curve at the point B=o the one axis is parallel to the axis of B. As the tracer follows the curve the frame is made to turn through an angle nO. At the same time the frame moves with the tracer in the direction of y. For a small motion the two wheels will then register just the components required, and during the continued motion of the tracer along the curve the wheels will add these components, and thus give the values of nA,, and nB,,. The factors I/yr and -1hr are taken account of in the graduation of the wheels. The readings have then to be divided by n to give the coefficients required. Coradi's realization of this idea will be understood from fig. 23. The frame PP' of the instrument rests on three rollers E, E', and D. The first two drive an axis with a disk C on it. It is placed parallel to the axis of x of the curve. The tracer is attached to a carriage WWwhich runs on the rail P. As it follows the curve this carriage moves through a distance x whilst the whole instrument runs forward through a distancey. The wheel C turns through an angle proportional, during each small motion, to dy. On it rests a glass sphere which will therefore also turn about its horizontal axis proportionally, to dy. The registering frame is suspended by aid of a spindle S, having a disk H. It is turned by aid of a wire connected with the carriage WW, and turns n times round as the tracer describes the whole length of the curve. The registering wheels R, R' rest against the glass sphere and give the values nA„ and nB,,. The value of n can be altered by changing the disk H into one of different diameter. It is also possible to mount on the same frame a number of spindles with registering wheels and glass spheres, each of the latter resting on a separate disk C. As many as five have been introduced. One guiding of the tracer over the curve gives then at once the ten coefficients A„ and B„ for n = I to 5. All the calculating machines and integrators considered so far have been kinematic. We have now to describe a most remarkable instrument based on the equilibrium of a rigid body under the action of springs. The body itself for rigidity's sake is made a hollow cylinder H, shown in fig. 24 in end view. It can turn about its axis, being supported on knife-edges O. To it springs are attached at the Michelson prolongation of a horizontal diameter; to the left a series 'n c of n small springs s, all alike, side by side at equal in-Stratton tervals at a distance a from the axis of the knife-edges; analyser. to the right a single spring Sat distance b. These springs are supposed to follow Hooke's law. If the elongation beyond the natural length of a spring is X, the force asserted by it is p=kX. Let for the position of equilibrium 1, L be respectively the elongation of a small and the large spring, k, K their constants, then nkla = KLb. The position now obtained will be called the normal one. Now let the top ends C of the small springs be raised through distances y2,...y,,• Then the body H will turn; B will move down through a distance z and A up through a distance bz. The new forces thus introduced will be in equilibrium if ak (1y—n1 z) =bKz. My— My l nb +a n (b +E) This shows that the displacement z of B is proportional to the sum of the displacements y of the tops of the small springs. The arrangement can therefore be used for the addition of a number of displacements. The instrument made has eighty small springs, and the authors state that from the experience gained there is no impossibility of increasing their number F C even to a thousand. The displacement z, which necessarily must be small, can be enlarged by aid of a lever OT'. To regulate the displacements y of, the points C (fig. 24) each spring is attached to a lever EC, fulcrum E. To this again a long rod FG is fixed by aid of a joint at F. The lower end of this rod rests on another lever GP, fulcrum N, at a changeable distance y"=NG from N. The elongation y of any spring s can thus be produced by a motion of P. If P be raised through a distance y', then the displacement y of C will be proportional to y'y"; it is, say, equal to µy'y" whereµ is the same for all springs. Now let the points C, and with it the springs s, the levers, &c., be numbered Co, C,, C2 . . There will be a zero-position for the points P all in a straight horizontal line. When in this position the points C will also be in a line, and this we take as axis of x. On it the points Co, C1, C2 . . . follow at equal distances, say each equal to h. The point Ck lies at the distance kh which gives the x of this point. Suppose now that the rods FG are all set at unit distance NG from N, and that the points P be raised so as to form points in a continuous curve y' =0(x), then the points C will lie in a curve y=µo(x). The area of this curve is µ f "4 (x)dx. Approximately this equals Mhy=hEy. Hence we have J (x)dx--My - -hz, where z is the displacement of the point B which can be measured. The curve y'=¢(x) may be supposed cut out as a templet. By putticig this under the points P the area of the curve is thus determined—the instrument is a simple integrator. The integral can be made more general by varying the distances NG = y". These can be set to form another curve y" = f (x). We have now yµy'y"=µf(x)4(x), and get as before J f(x)ct(x)dx=msµ z. These integrals are obtained by the addition of ordinates, and therefore by an approximate method. But the ordinates are numerous, there being 79 of them, and the results are in consequence very accurate. The displacement z of B is small, but it can be magnified by taking the reading of a point T' on the lever AB. The actual reading is done at point T connected with T' by a long vertical rod. At T either a scale can be placed or a drawing-board, on which a pen at T marks the displacement. If the points G are set so that the distances NG on the different levers are proportional to the terms of a numerical series uo+u,+u2+ and if all P be moved through the same distance, then z will be proportional to the sum: of this series up to 8o terms. We get an Addition Machine. The use of the machine; can, however, be still further extended. Let a templet with a curve y' =OW be set under each point P at right angles to the axis of x hence parallel to the plane of the figure. Let these templets form sections of a continuous surface, then each section parallel to the axis of x will form a curve like the old y' = ¢(x), but with a variable parameter f, or y'=¢(0, x). For each value of f the displacement of T will give the integral 1'= f f(x),b(fx)dx=F(0), where Y equals the displacement of T to some scale dependent on the constants of the instrument. If the whole block of templets he now pushed under the points P and if the drawing-board be moved at the same rate, then the pen T will draw the curve Y=F(f). The instrument now is an integraph giving" the value of a definite integral as function of a variable parameter. Having thus shown how the lever with its springs can be made to serve a variety of purposes, we return to the description of the actual instrument constructed. The machine serves first of all to sum up a series of harmonic motions or to draw the curve Y =a1 cos x+a2 cos 2x+a, cos 3x+ . . (2) The motion of the points PIP2 . . . is here made harmonic by aid of a series of excentric disks arranged so that for one revolution of the first the other disks complete 2, 3, . . revolutions. They are all driven by one handle. These disks take the place of the templets described before. The distances NG are made equal to the amplitudes al, a2, as, . . The drawing-board, moved forward by the turning of the handle, now receives a curve of which (2) is the equation. If all excentrics are turned through a right angle a sine-series can be added up. It is a remarkable fact that the same machine can be used as a harmonic analyser of a given curve. Let the curve to be analysed be set off along the levers NG so that in the old notation it is y"=f(x), whilst the curves y'=4,(xt) are replaced by the excentrics, hence f by the angle 0 through which the first excentric is turned, so that y k=cos IB. But kh=x and nh=,r, n being the number of springs s, and 7 taking the place of c. This makes ke = no.x. Hence our instrument draws a curve which gives the integral (r) in the form =2 -f of (x)cos (_Ox) dx as a function of 0. But this integral becomes the coefficient am in the cosine expansion if we make 0n/7r =m or 0 = m,r/n. The ordinates of the curve at the values 0=7/n, 27/n.. . give therefore all coefficients up to m=80. The curve shows at a glance which and how many of the coefficients are of importance. The instrument is described in Phil. Mag., vol. xlv., 1898. A number of curves drawn by it are given, and also examples of the analysis of curves for which the coefficients am are known. These indicate that a remarkable accuracy is obtained. (0. H.)
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