ATü'D THE LINES OF FORCE OF A CIRCULAR CURRENT. 11 



oj^ l8r\ >Uo) fUo) I 2 I 



7r.V(o|r,)\ 4 \ 'y,(o|rO 'XjolrJ // 



The expressions (17), (18), ;iiid (19) are of great practical importance, 

 as will be shown in another section. 



§ 8. Expression for -^ — . In addition to M, we shall have to 

 find -^r — , whicli represents the force acting between two coaxial coils 

 Since 



IM , r cos Odd 



/cos if do /.,^x 

 , (-20) 



we easily find that 



1 IM _ A'z /''A^(u) + B 



4 ./ ^V(w.)-'2j 



du 



(21) 



47Tax Is 4 J Af{u)—'IB 



A'z i 361 , , ,\ 



4 I 2(.9i — eaX«?! — e^) f 



Expressing e^, e^—e.,, e^—e^, by means of '^-functions, 



^z ^ ax \ 2 7Z- v;?o(o) fhioy } 



§ 9. 3/ expressed in q-series. — For reducing the '^-functions 

 in (17) and (18), we can conveniently make use of the expansions 

 given by Jacobi (Fundamenta Nova p. 104-105, Gesammelte Werke 

 Bd. 1. p. 161). As the result of expansion, we easily find that 



^ — = 4-7^(l+* + 3(/-4(7« + y(/'-12(/i° ) (23) 



47t^ax 



Putting 



8 (f*-4 f/ + 9 <?«- 12 (/°+ =£, 



