176 
MR. MURPHY ON THE ANALYSIS OF THE ROOTS OF EQUATIONS. 
E, = 
i) 
(\/ a — a 3 y) 
E 4 = -^^n(^ 
') 
A — »*(»- 1) 
Bj = — ”TI O'/ p-a-y y) 
C s = ^rn(^P-- 2 ^r) 
^6 — w 2 (a, _ 1) (\/ a \/ ^) 
B 6 — ~ «, ( fl 
1) 
(\/ a — u 3 \ / B) 
D 5 = 
c. 
-^r(^ 
-yi) 
e 5 
Therefore 
co (co 
5 
1) 
(\/ ft — \/ y) D 6 
i) 
co 4 (co — 1 ) 
y — h — h . A, B 2 C 3 B 4 E 4 
i3 — % = — h . A 2 B 3 C 4 Dj E 2 
a — y— h . A 3 B 4 C\ D 2 E 3 
a — (3 — — h . A 4 B 4 C 2 D 3 E 4 
(3 — y = /?, . A 5 B 5 C 5 D 5 E 5 
« — & = - A 6 . B ( ; .C 6 .Dg . E 6 _ 
(ct)‘ 
39. Now the conditions necessary for the evanescence of a" 1 are by art. 33. 
(a — h) ((3 — y) + (a — y) (f3 — &) = 0 
(a — y) (& — (3) + (a — (3) (h — y) = 0 
(a — (3) (y — i) + (« — &) (y — (3) = 0. 
Substitute the values of a — S, (3 — y, &c. above found, and including /* 2 in the nume- 
rical multiplier k of the whole, we shall have 
a "' — k' (A 5 A 6 . B 5 B 6 . C 5 C 6 . D 5 D 6 . E 5 Eg — A 2 A 3 . B 3 B 4 . C 4 C 4 . D 4 B 2 . E 2 E 3 ) 
X (A 5 Ag . B 5 Bg .C 5 Cg. D 5 Dg . E 5 Eg + A 4 A,. Bj B 2 . C 2 C 3 . D 3 D 4 . E 4 E 4 ) 
X ( A 2 A 3 . B 3 B 4 . C 4 C 4 . D 4 D 2 . E 2 E 3 + A 4 A 4 . B 4 B 2 . C 2 C 3 . D 3 D 4 . E 4 E 4 ). 
40. The condition that a” may vanish is 2 (a — (3) 2 = 0 by art. 33. Hence 
«" = V (A 2 B 2 C 3 2 D 4 2 E l 2 + A 2 2 B 3 2 C 4 2 Dj 2 E 2 2 + A 5 2 B 5 2 C 5 2 D 5 2 E 5 2 
+ A 3 2 B 4 2 C 4 2 D 2 2 E 3 2 + A 4 2 B 4 2 C 2 2 D 3 2 E 4 2 + Ag 2 Bg 2 C 6 2 Dg 2 Eg 2 ), 
k" being a numerical quantity. 
41. The terms which compose a! are indicated below by the indices which enter 
them placed between brackets. 
[5] =42 * 4 5 
[4, l] = — 5 2 Xj 4 x 2 
[3, 2] = — 10 2 j? 4 3j? 2 2 
