AND DOUBLE THETA-FUNCTIONS. 
941 
where X=abcdef, Y=a / b / c / d / e / f / ; which equations contain the constants ts, p, a, r, the 
values of which will be afterwards connected with the other constants. 
78. The c’s are expressed as functions of four quantities a, f3, y, S, and connected 
with each other, and with the constants (a, b, c, d, e,f) by the formulae 
c 2 _ 
0 = a 3 + /3 3 +y 3 +S'=m (J 2 y/bd, 
1 = 2(a/3+yS) = „ ce, 
2 = 2(ay+/3S) = „ y/cd, 
3 — 2 (aS + (3y) = „ \/be, 
4 = or—/3 3 +y 3 — S 3 = „ s/ac, 
6 = 2 (ay—/3S) = „ y/ ab, 
8 = a 3 +/3 3 —y 3 — 8 : = ,, y/bc, 
9 = 2(a/3—yS) = ,, y/de, 
12 = « 3 — /3~ — y 3 — S 3 = „ \/7cd, 
15 = 2(aS— /3y) = „ y/ae. 
It hence appears that we can form an arrangement 
r 2 
° 12? 
,,3 
0 n 
0 
(> 6 
. o, 
~ C 0 — 
a, 
c 
0 
O 
o 
_ C 4? 
C 3 
a', 
6 , c 
o 
0 
0 
// 
C 2? 
c h5> 
00 
O 
1 
a , 
b , c 
a system of coefficients in the trans¬ 
formation between two sets of rec¬ 
tangular coordinates. 
We have between the squares of these coefficients of transformation 6 + 9 equations 
that is 
cd +/r +c 3 =1, 
+2+/+ + + =!j 
a ''a_|_&''3+ c "3 =lj 
cd +a' 3 +«" 3 — 1, 
¥ +b'*+b"*=l, 
c 3 + c' 3 +c" 3 =l, 
tf+c z = a'*+a"\ 6' 3 + c 3 =a" 3 +« 3 , 6" 3 +c' ,3 =a 3 +«+ 
c 3 +a 3 =// 3 +?/' 3 , c*+a'*=b"*=b\ c" 2 -\-a" z =b 2 + 1/' 2 , 
a 3 +6 3 =c 3 +c" 3 , a' 2 +b /2 =c" 2 =c 2 , a" 2 -j-b" 2 =c z +c' 3 , 
c 4 
c 4 
c 4 
c 4. 
12 
+ 
1 
+ 6 
- 0 
9 
+ 
4 
+ 3 
— 0 
2 
+ 15 
+ 8 
- 0 
12 
+ 
9 
+ 2 
- 0 
1 
+ 
4 
+ 15 
— 0 
6 
+ 
3 
+ 8 
- 0 
1 
+ 
6 
— 9 
- 2 
6 
+ 12 
- 4 
-15 
12 
+ 
1 
- 3 
- 8 
4 
+ 
3 
_ 2 
-12 
3 
+ 
9 
-15 
- 1 
9 
+ 
4 
- 8 
- 6 
15 
+ 
8 
-12 
- 9 
8 
+ 
2 
- 1 
- 4 
2 
+ 15 
— 6 
- 3 
= 0 ; 
