254 Proceedings of Royal Society of Edinburgh. [sess. 
must vanish for all values of if/ and cf> , and that therefore a number 
of relations must exist between products of pairs of the coefficients. 
These relations Jacobi might have obtained by giving special values 
to if/ and <f > : for example, by putting if/ = 0 and if/ = 7r he might 
have obtained 
(a +y + y- i) + 2(aa' + /3/3' + yy - SS') + (a + ^ + / - f) = 0 
and 
(a +/3 2 + y- S 2 ) - 2(aa' + $8' + yy' - SS') + (a' 2 + ^ + / - S' 2 ) = 0 
and thence 
act! + /3f3' + yy — 88' = 0 
i 2 n 2 2 ,.2 . ,2 2 ,2 ~, 2 v 
and a + /3 + y — 8 = — (a + /3 +y — 8 ) . 
As a matter of fact, however, taking a hint from Gauss, he con- 
cludes that since 
cos 2 i fr + sin 2 ^ cos 2 cf> + sm 2 i[/ sin 2 <£ = 1 , 
the expression must he of the form 
k(cos 2 if/ + sin 2 i f/ cos 2 cf> + sin 2 i jr sin 2 <f> - 1) 
and that therefore by equalisation of coefficients 
a 2 
+ 
+ 
f 
- S 2 
= 
-k y 
2 
CL 
+ 
p 2 
+ 
,2 
r 
- S' 2 
= 
k, 
a 
+ 
/ 3" 2 
+ 
„% 
y 
- 8" 2 
= 
k, 
2 
a 
+ 
r 2 
+ 
»//2 
7 
- S"| 
= 
k , 
aa 
+ 
pp 
+ 
ti 
- 88' 
= 
0, 
rr 
aa 
+ PF 
+ 
rr 
77 
- 88" 
= 
0, 
nr 
aa 
+ ppr 
+ 
rrr 
77 
- 88'" 
= 
0, 
rr nr 
a a 
+ 
p'p" 
+ 
rr r»r 
7 7 
- 8"8'" 
= 
0, 
a!" a! 
+ 
+ 
y"Y 
- 8'"8' 
= 
0, 
r rr 
a a 
+ 
PP' 
+ 
/ // 
77 
- 8'8" 
: 
0, 
where k is arbitrary.* Again, since by making the substitution 
in the denominator 
G + G' cos 2 P + G" sin 2 P cos 2 0 + G'" sin 2 P sin 2 0 
The fact that these equations imply | afty'd"' | '= ±& 2 is not alluded to. 
