707 



by a process fundamentally that of Legendre, Second Supplement, 

 196. This is the equation by which E, and indirectly the third 

 integral II, is linked to the functions A9. 



6. We may further point out, as perhaps new, the developments of 

 A9 in the case when q is very near to 1 . Let r be related to b as 



q to c; then log-.log-=?r 2 . If log -=TT, and #=7rw, 



q r g. 



in which the double sign denotes two terms, which must both be in- 

 cluded. But besides, if the symbol p(r) stand for 



From these formulae not only all of Gudermann's developments 

 for calculating elliptic integrals in every case are deducible, but 

 others also, it seems, of a remarkable aspect, in the difficult case of 

 q and c being extremely near to 1 . 



We produce the two which seem to be simplest. Let B be to b 



what C is to c, and Tan x represent ^-^, where 2 Sin x stands for 



Cos a? 



c* e~ x and 2 Cos x for e x +6"-*. Then when c is very near to 1, 

 we compute G and thereby E from the series, 



The third elliptic integral is in the same case deduced from a series 

 of the form 



?~ 2 * ..;- (;-tan-Vtan; -Tan^H + 0-tan-iAaiy .Tan^^H 



VOL. IX. 3 B 



