114 PHILOSOPHICAL SOCIETY OF WASHINGTON. 
used if 6 and ¢ are not very different, for in that case the algorithm 
of the arithmetico-geometric mean converges equally fast in both 
directions. If either 6 or c is very near to a, the process may con- 
verge in one direction so slowly that the formula becomes nearly 
inapplicable. 
The fourth form may be transformed to a new formula, which is 
more convenient than any given. In (52g) place ¢ = 2” _], then, 
since ; 
P= 2-1 _J5 gy = I"? _J5 pe BMF _J wee 
on = athe Pn+1 — (2Pie Ws 8 “_2e 8 @ 
we have 
AL oi—1 Qn—2 
ple ae Ses gr aie eer 
i 1 
a eo ae Oe { _Nn41 wae (61) 
Gn +1 
But we have by (28) = sin(2(2h_)Pn+1—_)= i sin | 
or Qaiin? (OP. Dae 1 
Gn 
. | yaa 
Ser C2" Dp \/ (4 ae ) 
If we suppose a, = 6, = 6. within the precision of the compu- 
tation, ¢, will be very small, yet not zero. We have then 
i 1 
pecighec {@" Detite 
1 
= ttan #(_} + (2"_})n 41) 
Qn+1 
HE i Vi 1 + sin (2n Dna s 
ct wae 1 — sin (2? Dn 
Per rn 
Vig: h tye (1 +8) 
Tig yas 
Vityi (142) 
