352 Mr. J. W. L. Glaisher on [Nov. 20, 



(K + ^K') = (1 + h) \ 



IT- 



From this last equation the value of ©(^K + ^'K') may be deduced ; 

 but it is more convenient to avoid ambiguities of sign as follows. 

 In the formula 



Q(u + v)Q(u— v) =5-^^(1 — sn 2 % sn 2 i;), 

 put u=^K, v=^iK r , then 



= s rS»*'* (1 1 



since 1 + {2(1 + &) (1+ Tc r ) }K 



Also, in the formula 



eu 



put u= —^K—\iK.\ and 

 whence from (1) and (2) 



e 3 (4-k + i;k') = q-i—kW { (1 + #).* + *(i - ■ 



Extracting the square root, we have 



e(iK+i i -K')= 2 -^^[{2 i + (l + « ; ') 1 } i +i{2»-(l + //)*}*]; 



it is evident from the ^-series for (see § 8 of the present paper), 

 that the sign of the real or imaginary parts is positive as above. 



Since 4 (lK+iiKO = 2 -*?S*&'*(fc' + 



7T 2 



2K 2 



= q~ k ^—- k'Jc'l (cos 6 + i sin 6) , 



7T 2 



