110 PHILOSOPHICAL SOCIETY OF WASHINGTON. 
Solving (34,) for a, 5, ¢,, we have 
a,=a+b;b,=2Vab;¢=a=b 
In order to have again the convenient algorithm of the arith- 
metico-geometric mean, it is preferable to assume 
a, = 4a,= 3 (a+); b, = 4, = ab; 6, = be, = (@—b) (43) 
For the second step assume 
b 
tan (¢, — ¢) = = tan 9, (42,) 
1 
a,=t(a,+ b,); b= a,b; & = 2 (a, — 4) (43,) 
1 1 iL 
We have then ede 2a, (Q)n = Pa, Ce (44,) 
Continuing this process, which diminishes the modulus, and is 
therefore called descending the scale of moduli, while the above is 
called ascending, the a and 6 will rapidly approach their arithmetico- 
geometric mean, @~ = 6, while Fa Pin) tends towards a limit 
which I shall denote ¢. The limiting form of the integral is 
P(e) 
dg(ay _ Yoo 
Bey ibe \ 
and we have 
1 1 1 betel Dea . 
Ly — dq, Pn= ig, Pu)¥. = we tw ew 92 Ge (Yo), (=52 (44(~)) 
1 1 
If g = _|theni¢,= pe,=-+-pr 9m =--- = J 
and we have 
1 1 1 1 il 
A yet neh nee — 1h, (=F) 5m) 
This remarkable value for the complete integral was discovered 
by Gauss by means of a different transformation, known as Gauss’. 
This may be deduced as follows: Assume in place of (44(*)) the 
following series of relations 
1 1 1 
4 y= 1 Gon = EG On= +11 3z ode (= Foe Ge) 
