42] TAUTOCHRONOUS MOTION HARMONIC. 147 



Since this is to vanish for all values of the limit S we must have 



or 



18) 2 [F(s) - F(sJ] - sF' 0) + s Q F' ( % ) = 0, 



which is a differential equation for F(s). 

 Let us put y = F(s~), 



then 18) becomes 



19) s _ 22 , + c = 0, 



a linear, equation of the first order. An integrating factor is -g 



s 



multiplying by which the equation becomes 



Integrating, 



s 2 "^ 



20) y = 6s 2 + I 



From this we obtain 



so that the equation of tautochronous motion 14) must be 



oi\ ^ s 9/jc 



' ~dt* ' 



accordingly the motion must be harmonic, and evidently & must be 

 negative. 



We have seen that the cycloid is a tautochronous curve and 

 that a tautochronous vibration must be harmonic. 



By equation 6) the length of the arc of the cycloid measured 

 from s = is 



from which 2 



or inserting in equation 1), 



Differentiating, 



9A\ d * s = 9 s = i. 



/ /7* 2 _ / T~' I2~7 dt 4a 



