648 
PEOEESSOE STLYESTEE ON THE EEAL 
Then, since each letter occurs the same number of times (12) in each term, Z will be an 
invariant. 
(77) Again, suppose any two roots to become equal, say that e becomes d, then Z 
reduces to the single term £ (a, b, c) ^ ^ ; for any such factor as £(«, b , d ) will be 
accompanied with the factor ^ ^ which vanishes. 
If, further, we suppose any two of the letters a, b, c to become equal, then Z disap- 
pears entirely, since on that supposition £( a , b, c) vanishes. Hence Z is an invariant of 
the twelfth order, possessing the property of vanishing when the equation to which it 
belongs has two pairs of equal roots. Hence Z is of the form ^A+gJI), and it be- 
comes of importance to ascertain the value of the ratio -■ 
To do this let us suppose e=0, a=—b , c=—d. 
The ten terms in Z correspond to the following ten partitions : — 
(1) 
(2) 
abc 
abd 
de 
ce 
(5) 
abe 
cd 
(7) 
(8) 
ace 
bde 
bd 
ac 
(3) 
(4) 
acd 
bed 
be 
ae 
(6) 
cde 
ab 
(9) 
(10) 
ade 
bee 
be 
ad 
(78) The corresponding values of the terms will be 
4 a \a 2 -c 2 ) 2 . 16(aV)8 2 (a 2 — c 2 ) 4 ; 4a 2 (a 2 -c 2 ) 2 16a 2 c 2 (a 2 -c 2 ) 4 ; 4 c\a 2 -&) 2 . 1 6a 2 c\a 2 -c 2 ) 4 ; 
4c 2 (a 2 —c 2 ) 2 16a 2 c 2 (a 2 —c 2 ) 4 ; 4a 6 c 8 (a 2 —c 2 ) 2 ; 4 c 6 a 8 (a 2 —c 2 ) 2 ; (a — c) 2 256a 4 c 4 (a-\-cf; 
a 2 c 2 (a—c) 2 256a 4 e 4 .a 4 c 4 (a-\-c) 8 ; (a+c) 2 256a 4 c 4 (a—c) B ; (« + c) 2 2 5 6 a 4 c 4 ( a — cf. 
Collecting and simplifying these terms, and observing that 
(a—c) 2 (a+c) 8 -\-(a+c) 2 (a—c) 8 =(a 2 —c 2 )((a+c) 6 -\-(a—c) 6 ')=4(a 4 —c i Xa 4 -{-14a 2 c 2 -\-c 4 ), 
we find 
Z = 1 2 8(a 2 + c 2 )a 8 c 8 (a 2 — c 2 ) 6 + 4(a 2 + c 2 )a?c\a 2 —c 2 ) 8 
+ 1 02 4(<z 2 + c 2 ) (« 4 + 1 4a? c 2 + c 4 )(a 2 — c 2 ) 2 a 10 c'°. 
Let (a 2 -c 2 ) 2 =p, a 2 c 2 =g, and let Z,= firms' Then 
Z / = 1 6 3 8 4^2 3 + 1 0 2 4p 2 ^ 2 + 1 2 8^ 3 ^ + 4p 4 
= 2 yg 3 + 2 10 fq 2 + 2 7 fg + 2 2 f. 
