184 Mr Pocklington, On the Configuration of a pair [Feb. 26, 



we find 



^-± 



from (5) and (2) respectively. Also 



4"*" 4 



dn *( = 1 — I ¥ sin^ 2< = 1 — — + -^p cos 2w, 



'{u) = idull — Fsin^w— (l — -^U 



giving 



= — sm 2u, 

 4 



X = -^ sin 2u, 

 ttU 



2'= ^(1-2 + ^°^ 2") ■ 



and showing that the hollows are circles whose distance apart is 

 great compared with their radii. The velocity of translation of 



the vortex is V = (2a^ — 1 — 2a Va^ — l)U=^U,a value agree- 

 ing with that which we can obtain by elementary methods. 



The second tractable case is that in which k is nearly equal to 



2 

 unity. Here K = log r? , where k' is the modulus complementary 

 fc 



to k, and E=l, giving 



and thus 



dnu\ . 



2 

 Since the limits of dn u are 1 and 0, and log p is large, those 



of y are two small quantities, the larger of which, however, has a 



2 

 ratio to the smaller of the order J log -r, . 



