GENERAL CASE OP THE SIMPLE PENDULUM. 467 



therefore ^, -r-=/, — ; and by the principle of virtual ve- 



UL \jC Kit/ ^ 



w • 



locities, the sum of these forces, multiplied by the variations 

 of their directions, is equal to the action of gravity multiplied 

 by the variation of its direction. We have therefore 



d'x _. . d'i/ , d'z / . X 



Let the invariable length of the pendulum be denoted by 

 a, and let the common intersections of tiie three planes X, 

 y, and Z be in the vertical passing through the centre of 

 suspension at a distance equal to a below this point. We 

 shall then have 



^l(a—xy+tf+z'}=a, 

 and of which the variation is 



a a a 



an expression which the variations in the equation (i) must 

 satisfy. 



Let us therefore multiply tliis expression by an arbitrary 

 quantity 7' and add the result to the equation (i). Wc 

 shall then tind 



The variations ix, sy, n are independent in tliis ♦equation 

 by virtue of the arbitrary quantity T, and accordingly we 

 have 



