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
