MR. MURPHY ON THE ANALYSIS OF THE ROOTS OF EQUATIONS. 
175 
OO 1 
(2.) j 
(3.) ' 
(4.) 
(5.) 
( 6 .) 
HE.) 
(Xi - x 3 ) + (x 2 — x 3 ) + CO 2 (x 2 — x 4 ) . . 
(x 2 — x 4 ) + CO (x 4 — x 4 ) + co 2 (x 4 — x 3 ) . . 
(x 3 — X 4 ) + a (x 4 — x 4 ) + a 2 (x 4 — x 2 ) . . 
(x 4 - x 2 ) -f <y (x 3 — x 2 ) + CO 2 (x 3 - x 4 ) . . 
01 - X A ) + (<y 2 + co 3 ) (x 2 - x 3 ) . . 
02 — ^3) + (« 2 + a* 3 ) (x 4 — X l) • • 
The other thirty factors of « IV differ from these only in having cy 4 instead of co, thus 
changing the sign of (1.), (A.), to obtain the first of these factors, and writing it in an 
inverted order it gives 
u 2 {(x 5 - x 3 ) + *4 (x 4 - x 3 ) + co 3 (x 4 - x 2 )| 
differing from (4.), (A.) in having <w 4 for co, for the numerical factor u 2 maybe rejected 
since (cy 2 ) 30 = 1. 
The quantity a IV is the product of these sixty factors and a numerical constant k. 
38. Formation of the factors of a!”. 
The factors in the group A of the preceding article denote for abridgment by their 
number placed as a subindex to A, and so for all the others, thus by B 3 is meant the 
third factor in the group B. 
The quantity cc 1 " is composed of the three factors of ten dimensions each. Every 
such factor is the sum of two parts, each decomposable into ten simple factors. 
In the first pair of these simple factors x 4 does not enter, in the second pair x 2 is 
excluded, and so on. 
These three compound factors are found as follows : First, 
Ax = 
„(„_!) (V y- 
-v'*) 
A 2 — 
co (co - 1) ^ P 
-v'ii) 
b 2 = 
co 3 ( co — ] ) y 
— cy x/ S) 
B 3 = 
~ co 4 (co - 1) ft ' 
- cy 2 ^/ ci) 
C 3 = 
(«-i )(^y- 
&> 2 \/ &) 
e 4 — 
~ co 2 (co - 1) ^ ft ' 
— cy 4 -y/ 
d 4 = 
co 2 (co — 1) (n/ 7 
— co 3 })) 
Di = 
e 4 = 
w 4( w _ i) (s/ y 
— cy 4 ^ e)) 
e 2 = 
1 
1 “ 
"CO 
1 
- co 3 ^/l) 
B 
co (co 
5 
(x/ a - \/ 7) 
^4 — w (co — 1) ^ K ~ ^ 
4 = (V- J j U7 a - * 2 \/ y) 
a — co 
</ft) 
Cx 
D, 
co 4 (co 
irT) (\/ « - *y 7) 
co 3 (co — 1) 
zrfj (*7 a - « \/ 7 ) 
= 
^2 — w 4 ^ _ Jj(v^ a 0)2 V ft) 
Do 3 / 7T (\f & 
6 Or (w — 1 ) v v 
cy 3 x/(3) 
