Malet — On Hyper-elliptic Integrals. 
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LIY. — Some Theoeems m the Reduction of Hypee-elliptic In- 
TEGEALs. By John C. Malet, A. M. (Abstract.) 
[Read January 12, 1874.] 
In the first part of the present paper I prove a theorem due to Professor 
Gudan, and show that from this theorem the relations given by Jacobi 
among the moduli of hyper-elliptic integrals of the first class, for 
which they may be reduced to elliptic functions, and the other relations 
subsequently discovered, all follow by one uniform method ; in fact 
having proved one relation, the other two follow by an interchange of 
suffixes. 
I then generalize Professor Gordan's theorem, and from this gene- 
ralization prove that hyper-elliptic integrals with 2m - 3 moduli may 
be reduced to similar integrals with m - 1 ov m-2 moduli, according 
as m is even or odd, provided that certain equations connecting the 
moduli are true. 
There are three distinct set of equations among the moduli for 
which this theorem is true, and as in the case of the first class of 
hyper- elliptic integrals, being given one set of relations for which the 
theorem is true, the others follow by a mere interchange of suffixes. 
The last part of the paper is occupied with the consideration of a 
certain hyper-elliptic integral with five moduli, which I have shown 
may be expressed as an elliptic function. The formula I have given 
for its reduction leads to generalizations of many important theorems 
in the theory of elliptic functions. 
R. I. A. PROC. — VOL, 1,, SER. 11., SCIENCE. 
