1 4 m u 
t = ——— log ———— Ce U : p 
V1—a V1 —k? 27 3 (1—k ) log VTE 
because ME approaches 1, and 1 may be omitted by the side of 
2°—1 E 4 
RO SRF Now from (6) follows for large values of 
gere 
ih a Ee: 
fo et Ae ps ee Se 
zl ai ne Pe 
i 16 
7 = Paap: log Een 
1 32 (1—a)’ 
ee a ee, 
2V (1—a) ee en : i) abies 
4 
so that we obtain with Ian 
2 
sn p „(low temp.) . .- (7°) 
3 a 
i == ies 
32 (1—a)’ 
log | ———- p' 
a 
log qin) 
F = — —— F’, in which F’ refers to the integral with the complementary 
” 
modulus kh’ =W1-—k*, and gq’ is one of the auxiliary quantities q and q’, 
Ma (1 12 16 tc. 
introduced by Jacostr. From the relation yk’ — Pind orde ree) 
1+ 2q'+ 2q'*-}- 29" 4 etc. 
1+ q%+4" ene 
14 cia ip ) ‚from which 
1+29¢'+2q‘+... 
ele 
Dal 
SN me ne …. |. And from this follows: 
13 
‘ | 4 1 ' U 
ma en Benen 
1 
follows first of all — k= 9) 
i 
128 
1 1 9 
Through expansion into series #'' — „rf! ge Tae ae + BEE is easily 
1fgrc 
dw F' 1 
derived for Ff" = ,. So that from A= — —logq' 
V 1—k? sin? wp [4% 2 
0 
1 9 4 1 EBser 
finally ensues #’ = (: + mo Ee mn log oe ee dt pale 
A ll 
or approximated #’ — (2 + — ze) (tes 5 are | } the limiting value of which 
4 
is evidently log i 
We may still point out that the auxiliary quantity q always remains very small. 
q is=O0 for k’=0, but q’ is only 0,043 for k’ =1/.V2 (the same for k and q). 
It is to this fact that the exceedingly strong convergence of the Jacosi series for 
elliptical functions is owing. 
