A SYSTEM OF PARTICLES. 247 



Similarly, -j^ = - 2(^z,Ji . sin'' -^-a + 2 Fa A .sin*-— .a 



(1)- 

 In the same manner 



11. The integral of one of these equations is 



da da T,, ,„^ 



Wd7 = ^ (2)' 



a + a = Mi + N (3), 



but the circumstance that we have proceeded by supposing 



a = a cos {tit — kx), 

 a = «'cos {u't — kx + c), 

 gives u'a + u'-a = 0, by (1), 

 also ua sin {ut — kx) + u'd sin {u! t — kx + c) = — M. 

 Thus a + a = Mt + N becomes 



a -a = Mt + N, 



u' 



U-\ d'a 



I U'\ d'a 



V-¥^)df=''' 

 hence, u = u', 



a ^ — a , 



or, a cos c = — a, 

 a' sin c = ; 

 and c = TT, 

 a' = a. 



