102 PHILOSOPHICAL SOCIETY OF WASHINGTON. 
14rH MEETING. May 7, 1884. 
The Chairman presided. 
Nine members present. 
In the absence of the Secretary, the minutes were read by Mr. 
CHRISTIE. 
Mr. KumME Lt finished the paper begun by him at last meeting on 
THE QUADRIC TRANSFORMATION OF ELLIPTIC INTEGRALS, 
COMBINED WITH THE ALGORITHM OF THE 
ARITHMETICO-GEOMETRIC MEAN. 
[Abstract. ] 
The algorithm of the arithmetico-geometric mean, so remarkable 
for its symmetry and convenience, was first used by Gauss many 
years before the brilliant era of Abel and Jacobi. The form which 
the theory of elliptic functions assumed under the hands of these 
eminent geometers, though extremely beautiful, might be improved 
from a practical point of view by a combination with the Gaussian 
algorithm. In the attempt to do this, the defects of the usual nota- 
tion became very annoying, and gradually the new, simple, and 
consistent system of notations, as used in the following, resulted : 
I assume for the type of an integral of the first species, 
Y Y 
i adg ig adg 
V a@— sin’ V a cos *9 + 8 sin *9 
oO oO 
4 d 4 d 
V1—/;'sin’g V cos 29 + f sin *9 
oO oO 
For the inverse of this I write u_y = ¢. - (2) 
> 
By (1) we have the modulus 7 = < and the complementary modu- 
lus?=—-. The letters y and # are used throughout as symbols for 
a a 
required. 
c : P 
— and —, respectively, and are expressed in a, 6, and ¢ whenever 
