504 A MEMOIR ON THE SINGLE AND DOUBLE THETA-FUNCTIONS. [704 
where X = abcdef, Y = a / b / c / d / e / f / ; which equations contain the constants vr, p, er, t, the 
values of which will be afterwards connected with the other constants. 
78. The c’s are expressed as functions of four quantities a, /3, 7, 8, and connected 
with each other, and with the constants a, b, c, d, e, f, by the formulae 
c 2 
0 = a 2 +/3 2 + 7 2 + 8- = <o 0 2 ^bd, 
1 = 2 (a/3 -f 78) = „ v 7 ce, 
2 = 2 («7 + /38) = „ v/cd, 
3 = 2 (a8 + /37) = „ \/be, 
4 = a 2 -/3 2 + 7 2 -S 2 = „ '/ac, 
6 = 2 (a7 — /38) = „ v 7 ab, 
8 = a 2 -|- /3 2 — 7 2 — 8 2 = „ y/be, 
9 = 2(a/3 —78) = „ Vde, 
12 = a 2 — /3 2 — 7 2 — 8 2 = „ 
15 = 2(a8 — /87) = „ 
It hence appears that we can form an arrangement 
Cl2 2 } 
Cl 2 , 
C 6 2 
+ c 0 2 = 
a , 
b, 
c 
C 9 2 , 
-C4 2 , 
C3 2 
a', 
v, 
c' 
c 2 2 , 
- C 15 2 , 
-C 8 2 
a", 
b", 
c" 
a system of coefficients in the transformation between two sets of rectangular 
coordinates. 
We have, between the squares of these coefficients of transformation, a system of 
6 + 9 equations 
a. 2 + b 2 + c 2 = 1, 
a' 2 + b’ 2 + c' 2 = 1, 
a" 2 + b" 2 + c" 2 = 1, 
a 2 + a 2 + a" 2 = 1, 
b 2 + b' 2 + b" 2 = 1, 
c 2 + c /2 + c" 2 = 1, 
b 2 + c 2 = a' 2 + a" 2 , b' 2 + c' 2 = a" 2 + a 2 , b" 2 + c" 2 = a 2 + a' 2 , 
c 2 + a 2 = b' 2 + 6" 2 , c' 2 + a' 2 = 6" 2 + b 2 , c" 2 + a" 2 = 6 2 + 6' 2 , 
a 2 + b 2 = c /2 + c" 2 , a' 2 + 8' 2 = c" 2 + c 2 , a" 2 + b" 2 = c 2 + c 2 :