24 Allï. G. — II. NAGAOKA : THE INDUCTANCE 



K-E ^ 1 âj'(o) 1 ä,"(o) 



whence 



Expressing lliis in terms of q^s, we find for 8 



^ 3( i - 2(/ + 2(/' - 2f/')- 1 "^ 1 + (7- + ^'' + ^'^ 



8(g-V+9g^-...) ^.'-' 1 _ _4_J^ 

 ^ l-2^+27^-2(/«+-.. /r Î '^71 k' ^^ 



The above expression is applicable within wide limits of «, and 

 is rapidly convergent ; although q^"^ is retained, it is generally suffi- 

 cient for practical calculation from a=0 to « = 45° to suppress terms 

 beyond q\ For « = 45°, ^^ =0.00000349 and (7" =0.0000000065, so 

 that between the said limits 



8= 



2 {i-gfo^-a^U-^fci^i^'U-Al (21") 



3{l-2q+2qjr^^^^ ^ ^+ i-2</+2./ F } 3;:// ^"^^ ^ 



will give values accurate to six decimal places, which is super- 

 fluous in practice. Since -r- = cotga enters into the calculation 

 before finding q, it would be more convenient to retain it in this 

 form than to change it into a ^/-series. 



To find an expression of S in terms of q^, by which the 

 calculation of self-inductance for values of « from 45° to 90° may 

 be easily effected, we have to express E and K—E by means of 

 W3 and ^(o,ri)'s. 



Since E= U=(.;-i"'-^^) 



K-E= ^(.;_'"Li^;(£4) 



