On Lagrange's Equations applied to Fluid-Motion. 355 



where V denotes the potential energy due to the applied forces 

 which act on the bodies. 8x, By, 8z, as far as they apply to 

 the particles of the bodies, can, of course, be expressed in terms 

 of Sqij 8q 2 , &c., the variations of the generalized coordinates 

 determining the position of the bodies. And the same holds 

 true for the particles of the fluid, provided we suppose the 

 displacements irrotational. The above equation may therefore 

 be written 



dV dV 



Qifyi + Q2&/2 • • • + j- Bq x + j- 8q 2 . . . =0 ; 



and the equations of motion are 



dV dV 



To obtain Q x , suppose 



$q 2 =0, ^3=0, &c, 



Xdm(oc8x + y8y + z8z) 

 = %dm -j- {kSas + yhy + £8x) 



then 



Now if 



-Zdm(^8x + y^8y + z^Sz). 

 8x = a 1 8q l , ty = biSqi, 8z = Ci8q 1 , &c, 



when a ly b 1} c ± are functions of the <?'s and the coordinates 

 x, y, Zj then 



x = a 1 q 1 , y==biq u i=c 1 ^ 1 , &c; 



dec , dy ; 



dq{ dq{ 



. * . 2 dm -j- (cc8x + y§y + z8z) 

 a kj d ( . dx .dy x . dz\ ~ d (dT\ 



If now we are justified in assuming that 



= ^dm(x8x + '8y + z8 z ), 

 2A2 



