086 
PROFESSOR A. CAYLEY ON THE SINGLE 
necting the four theta-functions 4, 7, 8, 11 of a Gopel tetrad. And there is an equa¬ 
tion of the like form between the four functions of any other Gopel tetrad : for 
obtaining the actual equations some further investigation would be necessary. 
The x]j-expressions of the tlieta-functions. 
125. The various quadric relations between the tlieta-functions, admitting that 
they constitute a 13-fold relation, show that the theta-functions may be expressed as 
proportional to functions of two arbitrary parameters x, y ; and two of these functions 
being assumed at pleasure the others of them would be determinate ; we have of course 
(though it would not be easy to arrive at it in this manner) such a system in the 
foregoing expressions of the 16 functions in terms of x, y ; and conversely these 
expressions must satisfy identically the quadric relations between the theta-functions. 
126. To show that this is so as to the general form of the equations, consider first 
the ay-factors \fa, \/ab, &c. As regards the squared functions (\Zalf, we have for 
instance 
( \/aby= gn {abfc / d / e / +a / b / f cde -j- 2 v 7 X Y j , 
(\/cd) 3 =^{cdfa / b / e / -{-c / d / f abe-f- 2\/ XY j, 
each of these contains the same irrational part ^>\/XY, and the difference is 
therefore rational; and it is moreover integral, for we have 
(\faby — ( Veel)- - yo (abc / d / — ajpcd) (fe / — f e), 
where each factor divides by 0, and consequently the product by 0 2 ; 
fact 
= (<"-/) 
1, x+y, xy 
1, ci-\-h, ah 
L, c-\-d, cd 
the value is in 
a linear function of 1, xfy, xy ; and this is the case, as regards the difference of any 
two of the squares ( \/ah ) 9, , (f ac ) 2 , &c. ; hence selecting any one of these squares for 
instance (\/ def, any other of the squares is of the form 
X-\-p{x-\-y)~\-vxy-\-p(\/de)~; (p—l) 
and obviously, the other squares a) 2 , &c., are of the like form, the last coefficient p 
