Lines of Force of a Circular Current. 27 



Putting 3^'-47« + 978-I27I0+ . . . =e, 



we can briefly write 



^ -^=4.iTq\(\+e) (24) 



47r V ax 



Since e is a very small quantity we can, with tolerable 



accuracy, put 



M==167r' V«^y^ (^5) 



Expressing (19) in terms of ^1 



(l-yi4-4yi2-5ji^ + 6.^i^-4^i^ + 8^,''-%J + ..)}-4].(26) 



The above expression is useful when the coils are very near 

 each other. In such cases qi is a very small quantity, so 

 that by putting 



32^1' -40^iH48/^i'- 32^1^ + 64^1^-104^1^+ =ei 



4^ = .(i-2v,+iv+-y^t'^' "'K^J ( ' ■" ^^' 



(l-gi)+ei}-4]. . (27) 

 For A'>sin 70°. ^1^ is negligibly small, and 

 M 



^irV 



ax 



(l-ji + W)|-4]. . (28) 



Finally^ (22), gives 



qi{l + 20r/ + 225?' + 1840?° -^ 12120?' + ..) (29) 



BM 1927r2. 3,, , ^^2 



^2 x/ax 



It will not be altogether out of place to digress on the 

 practical utility of the several formulae above deduced. 



In the first place, there is no need of finding y connected 

 by the relation ^ = sin 7, /;^ = cos 7. Instead of finding 7 we 

 shall have to calculate 



^=2-4y-^K^)^ 



where l=^~ '^ ^' 



l+V^' 



