Lines of Force of a Circular Current. 25 



This expression coincides with that obtained from F. Neu- 

 mann's formula. 



5. Solid angle subtended hy a circle. — Denoting the integral 

 entering in (A^) by H, so that 



(a — x cos d)dd .^. 



Jo («' + ' 



x^ — 2aa; cos 6) V «^ + ^^ + ^^ — 2ax cos 6 

 and putting cos 6 = As-\-B, where 



\a.v/ ba.v 



we easily find that 



a = Az('''du + z(a'^.v')\'''~, , — f'' , , . ^c , (10) 



)^^ ■ ^J^_^a' + x'-2a.viAs+B)' 



where, following Weierstrass'' notation, 



r ds 



\/!S= \/^s^ — Q2S'-gz= \^[s — e^[s—e2){s—e-i) 



= sin 6 \l a^ -\-x^-\ z^ — lax cos 9 



2B 1-B 1 + B .... 



^i = -X' ^-^~A~' ^3= -^ ' ' (11) 



4:ax 



Let ^ 2(a^ + ^2)-^^ 



then : ^frt^^^s) 



(12) 



and 



n = ay,Az-^ii""-^4^,-'^du, (13) 



Jo>, 7W-7OO 



= ^^^+2i{^,«-a,,^(.)| (U) 



