Mr. POWER, ON A THEOREM IN FLUID MOTION. 457 



we obtain 



dp tttt , (u* + v* + w 2 \ du , dv j dw 7 

 ~f + dV-d( )--rf* + _rfy* -dz 



j (du dv\ (du dw\\ 



\ \dy dx) \dz dxl\ 



{(dv du\ (dv dw\\ 7 



U \Tx~ dy) +W {T*-dy)H 



j ( dw du \ ( dw _ dv\) , 



\ \dx dz) \dy dz) \ 



du 7 dv , dw 7 



= dt dx + di d y + dt d * 



(du dv\ . j , . (du dw\ , , 7 N 



+ \dy - dx) {vdx - ud ^ + te - a) {wdx - ud%) 



(dv dw\ , , 7 . 



+ [T*-dj) ( > 0d *- vd *>' 



Since p is a function of p depending on the nature of the fluid, 



— is a complete differential with respect to x, y, z, as well as the two 

 P 



remaining terms on the left-hand side of the above equation; conse- 

 quently, putting 



du dv n _ du dw _ dv dw 



dy dx' dz dx ' ^ ~ dz " dy ' 



du , dv y dw T 



Tt d * + di d * + -ai d *> 



+ a (vdx — udy) + (Z(wdx — udz) + y(wdy — vd%), 

 is a complete differential with respect to x, y, z. 



