. Graphical and mechanical computation . of the curve tobe analyzed, then evidently f Jo ydx = area OABCDBO; so that, if the tracing point of a planimeter is allowed to follow the curveOABCDBO, the integrating wheel will give the reading 2 xa0, from whichOo may be computed. * Proc. Roy. Soc, xxvii, 1878, p. 371; Kelvin and Taits Natural Philosophy. t Phil. Mag., xxxviii, 1894, p. no. X Phil. Mag., xxxviii, 1894, p. 121. § Phil. Mag., xxxix, 1895, p. 367; The Electrician, March 22, 1895. || Phil. Mag., xlv, 1898, p. 85. 1f Morin, Les Appareils dIntegration, 1913, p. 179. ** Elektrotech. Zeit.,
. Graphical and mechanical computation . of the curve tobe analyzed, then evidently f Jo ydx = area OABCDBO; so that, if the tracing point of a planimeter is allowed to follow the curveOABCDBO, the integrating wheel will give the reading 2 xa0, from whichOo may be computed. * Proc. Roy. Soc, xxvii, 1878, p. 371; Kelvin and Taits Natural Philosophy. t Phil. Mag., xxxviii, 1894, p. no. X Phil. Mag., xxxviii, 1894, p. 121. § Phil. Mag., xxxix, 1895, p. 367; The Electrician, March 22, 1895. || Phil. Mag., xlv, 1898, p. 85. 1f Morin, Les Appareils dIntegration, 1913, p. 179. ** Elektrotech. Zeit., xxxvi, 1909; Phys. Zeit., xi, 1910, p. The Electric Journal, xi, 1914, p. 91. XX Brief descriptions of all but the last of these may be found in Modern Instrumentsand Methods of Calculation, a handbook of the Napier Tercentenary Celebration, 1914.§§ For the principle of the planimeter and integrator, see pp. 246, 250. 204 EMPIRICAL FORMULAS — PERIODIC CURVES Chap. VII Integrating by parts, we may write -;— y sin kx ycoskx. WX i C2w . i P* ? -r- / sin kxdy = —-j— I slnkxdy,kit Jo kit Jo \w j P2* l P2* + T~ / coskxdy=—l coskxdy.) kirJo R-kJo Now if the planimeter carries twointegrating wheels whose axes make ateach instant angles kx and 71-/2 — kxwith the y-axis, and the point of inter-section of these axes is capable ofmoving parallel to the y-axis, then asthe tracer point passes around theboundary OABCDBO, these wheels give readings proportional to / sin kx dy and / sin ( kx J dy = I cos kx dy, from which the values of ak and &* can be found. In one form of the instrument the curve is drawn on a horizontalcylinder with the y-axis as one of the elements. A mechanism is attachedto a carriage which moves along a rail parallel to the axis, by means ofwhich a tracer point follows the curve while the cylinder rotates; themechanism allows the axes of the integrating wheels to be turned throughan angle kx while the cylinder ro-tates through an angle x. Coradi,the
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