108 PHILOSOPHICAL SOCIETY OF WASHINGTON. 
Solving for g, we have 
a sin 2¢’ a tans’ 
baa arn + a cos 2¢’ ~ a’ — U’ tan 79’ (30) 
one 2a sin ¢’ cos ¢’ __ asin ¢’ cos ¢’ (31) 
Pe Ya + 2ac cos 2¢ +e a ag’ 
e+acos2¢’ 1 b’ 
cos ¢ = ret as (da é—z3) (82) 
et ae bv’ 
ag=—(a A¢g +z9) (33) 
where we have placed 
t(ate)=a;t(a—ec) =U; 
_Vac=¢; Va? —c? sin?’ =a ag (34) 
From (82) and (33) follows 
ado’ =} (ad¢-+c cos ¢) (35) 
b’ 
ag TGR Ps core) (36) 
d dy’ 
and (29) becomes rye =7ag (37) 
1 1, 
the integral is 7 %7=a gy (38) 
‘The first and third formula of (34) give the first step in the 
algorithm of the arithmetico-geometric mean, and the first two fol- 
low from (35) and (86) by placing g = 0 = ¢’, 2. e., they are rela- 
tions at the lower limit of the integrals, corresponding to (35) 
and (86). 
/ 
Assuming sin (2¢” — ¢’) = — sin g’ (28’) 
oe’ = 3 (a’ + d); jess } (a’ ee c); eee Vad (34’) 
1 1 / 1 "et 
then we have a ay ety = Tey (38’) 
Proceeding in this manner the amplitudes will very rapidly 
reach a limit ¢‘~), while simultaneously a and ¢ tend to become 
equal to their common limit, the arithmetico-geometric mean of a 
and ec. Gauss, when investigating its functional properties, denotes 
