256 



MATHEMATICAL FORMULA AND ELLIPTIC FUNCTIONS 



'. s = _ ( x _ e)H + zs eK 



/*i s - si ds C 52 s 2 - s y 5 - 



f S-5 2 _rff_ - ^ pl-S 

 J -vAT^ls" VS J S - 5 3 



f *-ft _^ = A2 - S 3 VS1 ~ S3 ^ = (][ _ ^ K _ #) + zs eK 



J Vsi - s 3 A/S J s - s * Vs 



the integrals being o at the upper limit, 5 = , or at the lower limit, 5 

 where e = o and zs eK = . 

 So also, 



5 5 - 5 2 A/Si - S 3 , __ C S> ^_Sl - 5 (/5 eH + Zn eA 



5 53 \/5 ^ S3> s A 



/ / f* o c 



/ 5 5i V Si S3 , _ I 52 5 



J S S3 ^/S " A/Si J 



'*/ s 5 3 y'.s* / \/si - 



Similarly, for the variable a in the regions 



S negative, and 



- 5 3 V^ (i - e)H -zn d 

 ds e(H - K f *K) + zn eK 



- H) - zn e^T 



zn 



- 5 2 si - <r 



5! -(7 



/" (7-5 2 <frr rsg- 



J VsT^s" V" 1 ^ J ^ - 



/0--53 </(T A 2 - 



/s^l" /^S " / si - 



(i - j)(H' - K f *K') - znfK' 



f" S ^Vs^ dff = fJ^d m (I .^^ _ fll) + KfK , 

 Js z Si a -\/ z, t/o- vsi 53 v 2/ 



^r*n 



J 5i- 



- <7 V Si - S 3 



/5 2 (7 da 

 ^n 



Si - S 3 V - 

 (7 rf(7 



/ I _^V L ^^ = r^^ 



/ 5i <7 y' _ 2 / \/Si 5s \/ S 



these last three integrals being infinite at the upper limit, <7 = 5i, or lower limit 

 <7 = oc^ where / = o, zs /A' = oo . 



Putting e - i or / = i any of these forms will give the complete E. I. II, 

 noticing that zn K' and zs K f are zero. 



