50] GYEOSCOPIC FORCES. 187 



and the part contributed by the particle A to the kinetic energy is 



The opposite particle C, for which x 2 , y 2 , xy have the same values, 

 contributes the same amount. The other pair of particles, for which 

 the values of x 2 , y 2 are respectively those of y 2 , x 2 , for the first pair, 

 and the values of xy the negatives of the values for the first pair, 

 consequently contributes an amount of energy which, added to that 

 already found, makes the terms in xy disappear, and replaces each 

 term in x 2 , y 2 , by the same term with I 2 written in the place of x 2 

 or y 2 . Neglecting then | 2 , if, we have finally 



176) y = ' 



We accordingly see that cp is a cyclic coordinate for the system, so 

 that if the system is spinning without any force tending to change qp, 

 we are dealing with a case of the example in 49. We have, pro- 

 ceeding as there, 



177) 

 and eliminating qp f , 



from which we form 



In order to form the diiferential equations for the motion of |, 77, 

 we have by differentiation 



179) 



and neglecting the squares and products of the small quantities |, 

 and ', ?/, which are small at the same time, 



w 



a* 



ai 



