JBBMMMIIb 
[310 
310] NOTE ON MR JERRARD’S RESEARCHES ON THE EQUATION &C. 51 
EQUATION 
214] 
bution whatever, 
pical. Thus all 
y — z — w)- and 
ry of equations, 
ny homotypical 
typical function 
1 of the given 
y on the Reso- 
of an equation 
er in W; and 
n is a rational 
to a class of 
has shown, be 
ice I infer the 
ite combination 
«quadratic equation 
and in either case 
n, the homotypical 
hen, given a + b = 8, 
at here ab is deter- 
The above property of rational expressibility, if true for W, will be true for any 
function homotypical with W; and conversely. I proceed to inquire into the form of 
the function W. 
The function W is derived from the function P, which denotes any one of the 
quantities p x , p 2 , p 3 . And if x x , x 2 , x 3 , x 4 , x 5 are the roots of the given equation of 
the fifth order, and if a, /3, 7, 8, e represent in an undetermined or arbitrary order of 
succession the five indices 1, 2, 3, 4, 5, and if 4 denote an imaginary fifth root of 
unity (I conform myself to Mr Jerrard’s notation), then p x , p 2 , p 3 , and the other 
auxiliary quantities t, u, are obtained from the system of equations: 
x a 3 +p 1 x a 2 +p 2 x a +p 3 = t + u, 
ocp 3 +p x Xf +p 2 Xp +p 3 — 1.1 + thi, 
x y 3 + p x Xy- + p. 2 x y +p 3 = i 2 t -f vhi, 
xi + p x x£ +p 2 x 8 + p 3 = i 3 t + ihi, 
x 3 + p x x 2 + p 2 x e 4- p 3 — lH + i u. 
If from these equations we seek for the values of p x , p 2 , p 3 , t, u, we have 
1 : p x : p 2 : p 3 : -t : -u=U 1 : IT, : II 3 : TT 4 : IT 5 : II 6 , 
where IT, II 2 ,. . denote the determinants formed out of the matrix 
rp 3 
*As a ? 
rp 2 
tU a ; 
%a, 
1, 
1, 
1 
Xp 3 , 
Xp, 
1, 
4 4 
rp 3 
tty , 
rp 2 
dry y 
Xy , 
1, 
u 
4 3 
x s 3 , 
Xf, 
x s , 
1, 
X, 
4 2 
X 3 , 
rp 2 
0/ € , 
1, 
4 
i.e., denoting the columns of this matrix by 1, 2, 3, 4, 5, 6, we have Eh = 23456, 
Eh = — 34561, IT 3 = 45612, &c. In particular, the value of Eh is 
rp 2 
Xa. > 
1, 
1, 
1 
Xp 2 , 
Xp , 
1, 
c 
4 4 
Xy~ , 
Xy, 
1, 
r, 
4 s 
X S 2 , 
x$, 
1, 
l :i , 
4 2 
rp 2 
d/ e y 
x e , 
1, 
4 
and developing, and putting for shortness {a/3} =x a xp (x a — xp), &c., we have 
Eh = ({a/3} + {/37} + {7S} 4- {Se} 4- {ea}) (- 21+ 4 2 - 4 3 4- 24 4 ) 
+ ({07} + {76} + {e/3} 4- {/38} + {¿¡a}) (4- 4 4- 2r — 2t? — 2i 4 ); 
7—2