384 On Helmholtz's Memoir on the Conservation of Force. 



the first will be 



dv 

 = D i7j ~ dt dx +p 2 v l dt —p l v l dt, 



dv 

 Q = ~Dv_ 1 —-=—dtdx+pv- l dt—p- 1 v- l dt. 



Hence, corresponding to the different elements of the entire 

 wave, we shall have a series of equations, such as 



0=&c, 



= Dv_ 2 — j—dtdx+p^ l v_ 2 dt—p_2 v -2dt i 



0='Dv_ l —~ dtdx+pv^dt—p-iV-idt, 



dv 

 0=D# -j-dtdx+PiVdt—pvdt, 



dvt 

 0=Dv! -jjdtdx+p 2 v x dt — pv^dt, 



dv 

 = D v 2 -jjdtdx +p 3 v 9 dt —j9 2 v 2 dt, 



&c. = &c. 



Taking the sum of these, observing that the first and last terms 

 will vanish, the particle-velocity at the extremities being either 

 zero or infinitely small, we get 



0=% (j)vj dt dx) +2j)(v__ 1 --v)dt 

 ==2 (^v-fi dtdxj +2,p(v— -j-dx—v)dt 

 =X yDv -j-dt dx) — 2p -7- dx dt 



-i.-s-i'**" 



the integration being extended over the entire wave, or 



°=i m i dt+ i v i dt (a) 



(since I »-r-=/w — I v £-. the first term on the right-hand 



J/ dx X dx 



side of which equation vanishes when the integration is extended 

 over the entire wave, the velocity vanishing at the extremities of 

 the wave). 



