or, if we herein restore the c’s in place of the rectangular coefficients, this is 
x 4 + y 4 + z 4 + IU 4 
o c ° c 3 c iiA5 c» y _ q 
+ Z r 2 r 2 r 2 c '2 r 2 r 2 X H ZW U > 
t/j l/ 2 W C/6 ^9 
which is the equation of the 16-nodal quartic surface. 
Substituting for x, y, z, w their values A 4 , ^ 7 , ^- 8 , *& n (u), we have the equation 
connecting the four theta-functions 4, 7, 8, 11 of a Gopel tetrad. And there is an 
equation 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 xy-expressions of the theta-functions. 
125. The various quadric relations between the theta-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, fab, &c. As regards the squared functions (fab) 2 , we have for 
instance 
(fab) 2 = ~ {abfc / d / e / + a.b^cde + 2 fXY}, 
(f cd) 2 = ~ {cdfa / b / e / + c / d / f / abe -f 2 fXY] ; 
each of these contains the same irrational part ^fXY, and the difference is therefore 
rational : and it is moreover integral, for we have 
(f ab) 2 — (f cd) 2 = (abc / d / — a b / cd) (fe 7 — f e), 
where each factor divides by 6, and consequently the product by 6 2 \ the value is in 
fact 
= (e~f) 1, 
1, 
a + y, 
xy 
a + b, 
ab 
c +d, 
cd