^56 Mr. POWER, ON A THEOREM IN FLUID MOTION. 



the present communication to shew that even in cases where u, v, w, 

 do not admit of developement in the supposed form, the theorem is 

 nevertheless true. 



The general equation of the motion of fluids is 



which, since 



d , N du du du du 



dt dt dx dy dz 



d . ! c?« r/r rf» rf» 



rf# «?£ d# dy d% ' 



d dw dw dw dw 



dt dt dx dy d% ' 



becomes 



dp ,„ (du du du du ) , 



- + d V = {-j- + -_— u + -j- v + -j- w ) dx 



p [dt dx dy dz ) 



[dv dv dv dv 1 , 



+ \dt + ~dx u + d7j vJr i[; w ) d y 



{dw dw dw dw \ , 



di + dx^ U+ ch, V + dz- W ) d% > 



where dV ' — Xdx + Ydy + Zd%. 



If we subtract the identity 



, (u* + v 2 + uf\ i du dv dw) , 



d \ sT — } = \ u di + V dx + w dx) dx 



I du dv dw\ 7 



+ { U dy + V di + W dy) d y 



( du dv dw\ 7 



+ { U dz +V dz +W dz) d *> 



