468 GENEBA.I. CASE OF THE 



df a ^ ' 



df a 



df fl ' 



which are easily verified by observing that the first terms 

 are the components of the inertia of the oscillating point in 

 the directions of a:, ?/, Z; the second terms, the components 

 of the tension of the string in the same directions ; and g the 

 effect of gravity in the vertical direction x. 



Multiplying these equations respectively by do:, dz/, d^;, 

 adding the results, and observing that by virtue of the equa^ 

 tion (3) we have 



X- 



— da:+^d2/ +— d2;=0, 



a a a 



we shall get 



d'a;, d'«, d'z, , 

 d^^'^+dP^Z+^dfe^d^, 



of which the integral is 



At' 



2g{b-x). (3) 



Again, multiplying the second and third of the preceding 

 equations by z and y, and taking the difference of the results, 

 we shall find 



d^w d^z n 

 Z - — II — =0, 



d^^ -^df ' 

 of which the integral is 



