1901-2.] Dr Muir on the Theory of Orthogonants. 257 
collection along with the corresponding one of the latter, four new 
sets of four are deduced, which on rearrangement stand thus : — 
A(a + G') 
+ 
Alb' 
+ 
Al'b" 
+ 
Al"b"' 
== 
o, 
A V 
+ 
A \a 
-G') + 
A" e" 
+ 
Al"e 
= 
0, 
Ah" 
+ 
Ale" 
4- 
A!' (a 
-G) 
+ 
Al"c 
= 
0, 
AV" 
+ 
A'cl' 
+ 
AV 
+ 
A!" (a'" - G') 
= 
0; 
B(« + G") 
+ 
B 'b' 
+ 
B "b" 
+ 
B '"b"' 
= 
0, 
B b' 
+ 
B \a 
- Gy") + 
BV" 
+ 
B'V 
= 
0, 
B b" 
+ 
BV" 
+ 
B"(a" 
-G") 
+ 
B"V 
= 
0, 
m" 
+ 
B'c" 
+ 
B"c' 
+ 
B "\al" - G") 
= 
0; 
C(a + G"') 
+ 
C 'b' 
+ 
G"b" 
+ 
C '"V" 
= 
0, 
C b' 
+ 
G\a 
- G'") + 
G"c" 
+ 
G"'e 
= 
0, 
C b" 
+ 
G'c" 
+ 
G"(a" 
- G"') 
+ 
G!"e 
= 
0, 
c r 
+ 
G'c 
+ 
G V 
+ 
C'"(a" - G'") 
= 
0; 
D(a-G) 
+ 
D7/ 
+ 
D "b" 
+ 
D "'b'" 
= 
0, 
m 
+ 
D (cl + G) + 
DV" 
+ 
B"'e' 
= 
0, 
m" 
+ 
D V" 
+ 
D "(cl" + G) 
+ 
D'"c' 
= 
0, 
+ 
D'c" 
+ 
D V 
+ 
D '"(a" + G) 
= 
0. 
The elimination of A , A' , A" , A'" from the first set of four ; 
B , B' , B" , B"' from the second set of four ; and so on ; gives rise 
to four equations, the first of which is a quartic in G' , and the 
second, third, and fourth differ from the first merely in having 
G" , G"' , - G in place of G\ This, of course, is the same as saying 
that G' , G" , G'" , — G are the roots of a certain quartic in x, which 
would nowadays be written 
a - x b' b" 
b a +x c 
irr ttt ir . 
b c a +x 
i /// tr f 
0 c c 
but which Jacobi writes in the form 
b"' 
c 
cl 
a" + x 
= 0, 
(a - r)(al + x)(a" + x)(ci" + x) 
2 2 2 
(a - x)(a + x)c - (a - x)(a" + x)c" - (a — x)(a" + x)c"“ ) 
9 > 
- ( a " + x)(a" + x)b' u - (a" + x)(a + x)b" 2 - (a + x)(a" + x)b'"* ) 
+ 2 cc'c"(a - x) + 2 cb"b'”(a + x) + 2 cb'"b\a" + x) + 2 c"b'b"{a!" + x) 
V" 2 - 2 b'V'cc" - 2 b"b"'c"c'"\ - 2 b'"b'c'"c , 
17 
, ,2 /2 ,„2 f/ 2 7 n ,2 
+b c +b c +b 
PROC. ROY. SOC. EDIN. — YOL. XXIY. 
