COEFFICIENTS OF SOLENOIDS. 23 



Starting from the well-known expressions^^' 



K{K-E) = 27z'q{\ + 2q + ^q^-V^q^+Qq^-^'èq'+ ) 



KE = ^ {l-\-Sq-{l-q^+Aq'-5c/'+ )} 



¥ K= ii(l-4(i4-4(/+42^-8(?-^ + V- )} 



h' K" = 4-\^(/ + 4(?H Gr/ + 8g^ + ) 



we easily find by (I) 



8 = l--^^, +2^ + r25^ + 44r/ + 116^H2G0r/ + 5769«+ (21) 



The limits of «, within which the above series can be convenient- 

 ly used are wider than the expansions already given ; for «=45° 

 5765*' = 0.00000375, so that the values of S are right to six 

 decimal places for the above argument. By the way, it may be 

 noted that for the same value of «, the term affected with l-^' in 

 (20) =0.0001324, and the convergency of (20) is slower than that 

 of (21). 



§ 21. In order to arrive at expressions, which would give 

 more exact values of 8, it is necessary to transform formula (17) 

 into ??-functions. 



Q. 1 ^"2(0) 1 ^"0(0) 



' 4^1 äjo) 4i(0i ä^,{o) 



E = -^={-f], + e,co,) 



K - E= ^ (y], + e,cü,) and // - V^^l-ll 



we find ^ - - ___1__ jVM- L_ m± 



(1) Jacobi, Fiindamenta Nova, 105 ; Werke. 1, p. IGl. 



