VvKSK\i~ J /)/j/inff ions of I)('s.s(>/'s Funcfions fo Pf/z/.s/cs 



(il 



Rotatory Motion of a Fhiid, fn'ct/ou hciny taken account of. 



In the problems wc sliall consider, we sliall assume the cylindrical 

 rotation form nice 



W = - coy, V = WXy 10 = 0, 



where w, the angular rotation, is a function of the distance r from the 

 axis of the cylinder. The equations of motion are now 

 dp • dp • 



/ dv du\ ^ ( dv du\ ^ . 



du 

 dt 



du 



+ w — + 2 

 dx 



du 

 dy 



du 

 " di 



- ii)-X, 



dv 

 di 



dv 

 dx 



dv 

 dy 



dv 

 " di 



- <^V- 



du 



d dv 



d 



du 





dx dy) [dx d'yj 



C being the molecular rotation 

 Now, 



Hence, 



dx dy dx dt dy di dt ^ 



an expression, we may observe, arrived at without assuming sniallm ss 



of such terms as u ^ in comparison with • 

 dx ^ dt 



Assume, now, t, = e-'^'(f>{r) ; then, writing — = a^, Ave have 



dr r dr 



We shall then obtain solutions of vortex problems by taking 



the system of m being determined by the particular conditions of the 

 problem considered. The rotation w is connected with tlic moKfiihir 

 rotation ^ by 



du^ \ d ^ 

 2C = 2u) + r — = - - (wr-). 

 d r r dr 



