﻿458 Mr. G. B. Jeffery on the Tico-Dimensional 



If a = this gives 



yfr=:—bvij + Ae bx + Bj/ + C l r, . . . (I.) 

 while if a is not zero, a shift of origin gives 



2vaxy + A 1 1 1 <? flx2 (da?) 3 , • • ( IL ) 



«-c 



(I.) and (II.) are the only distinct solutions for which the 

 lines of constant vorticity are a set of parallel straight lines. 

 In the second case, when /e = l and X=l 



a -H'/3= log(*+iy). 



Hence, if r, 6 be polar coordinates 



a= log r, fi = 0, 



and 



^. = _^(9al0gr + 6) + 1—1 ^^ | e «(logr)2+6(logr)+ C JL 



This also leads to two distinct solutions according as a is or 

 is not zero. 

 Ifa=0 



^ = _6H9 + A^+ 2 + Br 2 -f Clogr [bzfzQ or - 2 )~l (III) 

 = 2z;(9 + A(logr) 2 + Bv' 2 + Clogr (6= -2), J 



while if a is not zero, a change in the scale of r gives 



f=-2va0\og r + A (-[rarfa 10 ^^. . (IV.) 



(III.), (IV.) are the only distinct solutions for which the 

 lines of constant vorticity are a set of concentric circles. 

 We have thus obtained all the solutions for which the lines 

 of constant vorticity are the equipotential lines in free space 

 of some possible distribution of matter. 



§ 2. Solutions for which the stream-lines are possible 

 equijiotential lines. 



The characteristic property of this type of solution is that 

 it is possible to choose a set of curvilinear coordinates a, /? 

 so that i/r = /■'(«). It has not been found possible to solve 

 this case with the generality of the previous section. Sub- 

 stituting in (2) we have 



