26 Prof. H. Nagaoka on the Potential and 



Expressing eiw^ and r)i(Oi by means of ^-functions, >ve 

 obtain'" 



M= 2V«:rfV(0) V(0)> ^7. 



V(0)U>'(0) "3,(0) f' • • • V ; 



«"• M : 2v/^rV(0) V(0)1 (18) 



V(0) 1^0(0) "^ MO)/' ■ ■ • ^ ' 



Similarly (B2") can be written 



8v/«^ r , Kg,)/ V(0/T,) V'(OMU nqx 

 ~ 'rVWrT) l'^ --T U(0/t,) +33(0/tO ; J ' ^ ^ ^ 



where _ _ toi 

 ^~ (W3* 



The expressions (17), (18), and (19) are of great practical 

 importance, as will be shown in another section. 



7. Expression for -z^ In addition to M, we shall have 



to find T^— , which represents the force acting between two 



coaxial coils. 



Since BM . ("^ cos OdQ ,-,^, 



— — =4:7raxz\ — -, . (20) 



^^ J^ {a^ + x^ + z''-2aa: cos d)^ 



we easily find that 



1 BM _ A^ r^3 A7(^)-fB 



which, expressed in ^-functions, gives 



8. M expressed in q-series. — For reducing the d-functions 

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

 pressions given bv Jacobi *. As the result of expansion we 

 find 



M — 



'-^^^=^7rqHl-hSg^-4.q'-{-9q'-12g''-h...). (23) 

 47r V «^ 



* Jacobi, iFujidamenta Nova, pp. 104, lOo ; Gesammelte Werke, i. p. 16L 



