OF INCOMPRESSIBLE FLUIDS. 451 



or, using (17). 



du. du, s 



dr d% r 

 whence we easily see that u. = r is one such factor. 



Let then 



dll= rsdz — rwdr, 



1 dU 1 dU 



so that * = j— , w = j— . 



r d% r dr 



The equation which U is to satisfy will be got by expressing s 

 and w in terms of U, and substituting in (19) in the general case, or 

 by substituting in (18), in the case where udx + vdy + wd% is an exact 

 differential. 



In the latter case the equation which V is to satisfy is 



rf^ + dr* r dr ~° W 



In the general case, the equation is what I shall write 



tdUd^ dU d\ LI /d*U.d*U ldU\\ ,„. 



\d» dr dr ds) {? 1 3? + dr* r dr)) * h 



The value of p is given by the equation 



p [[( ds ds\ , ( dtv dw\ , \ 



- P - - J {(* 3? + w 55 ) dr + (* dv + w an) dx 1 • 



, J , „ ,N ^* I ^^ 7 ^8 j dtD j 



Now \ d (f + w*) = * -j- dr + w^r- d% + s-j-d% + w-j- dr; 



and therefore 



ds ds\ j ( dw , dw\ 



i ds ds\ , . l dw , dw\ , 



{ s Tr + W di) dr + i S dr^ + W di) dz 



= £d(s 8 + w % ) + -T- (wdr - sd%) + -j- {sd% - wdr) 



