56 ROYAL SOCIETY OF CANADA 



and that the values of the constants are given by a series of definite 

 integrals, viz.: 



A — —- f f (X) dx 



IT 



f {x) COS mx dx 



27r 



2 /^ZTT 



B^ = — / f (x') sin mx dx. 



And it can also be shown that no other expansion in a series of sines 

 and cosines can represent the function. 



If the function f^z) is not finite everywhere the series may be divei*- 

 gent, but if it is finite the series must be at least semi-convergent, while 

 if discontinuities occur it can be shown that the series will still represent 

 the function, except at the discontinuity, when the sum of the series for 

 this value of x is the mean of the arithmetic values off{x) at that place. 

 In general, therefore, a curve representing displacements, velocities and 

 acclerations can be resolved into a series of sines or cosines, and since 

 integration of a series renders it more convergent than before, it is 

 possible, starting from a velocity time curve, or an acclei'ation time curve, 

 to obtain a corresponding displacement time curve, from which a cam 

 can be ccnstructed giving p rede ter n>ined velocities and acclerations. 



As a simple example let it be required to determine the form of a 

 cam, which, for one half its stroke, gives a uniformly increasing velocity 

 to the follower, and for the other half a uniformly decreasing velocity. 

 It can easily be shown that a Fourier series representing the given 

 motion is 



8 F (■ 1 (-1)" 



V= — -^ ■{ sin Out — — sin 3 oot 4- . . -\ ~ sin (2n-l) GJt-\- 



TT^ { 3- ' ' {2n-iy ^ ^ 



where oa = -^ V 



and this is obviously convergent, since the coefficients are so, and the sine 

 terms can only vary between db 1. If now we integrate we obtain 



S-{-S = —^Z^fcosœt —\..cos3 œt + ^T^]l, cos (2n -1) cot ■{- . .\ 



a still more converging series, and hence the displacement can be deter- 

 mined and the cam drawn. 



As another example, let us take an acclei-ation curve in which a 

 uniform positive accleration is impressed upon the follower for one half 

 a revolution, followed by a uniform negative accleration of the same 

 numerical value. For this case 



4:a ( 1 1 



a = ■ — ° ^ sin cot -\- ^ in 3 Out -\- . . -\- .7-3 sm (2n-l) œt -\- . . . . V 



