376 
on tschirnhausen’s transformation. 
[275 
and we have thence for the value of y, 
у = {ax + b) В + (ax* + 4bx + 3c) G + (ax 3 + 4bx 2 + 6cx + 3d) D ; 
so that for this value of у the function of у which equated to zero gives the trans 
formed equation will be an invariant of the two quantics. It is proper to notice that 
in the last-mentioned expression for y, all the coefficients except those of the term in 
x°, or bB + 3cG + SdB, are those of the binomial (1, l) 4 , whereas the excepted coefficients 
are those of the binomial (1, l) 3 ; this suffices to show what the expression for у is 
in the general case. 
I have in the two papers, “ Note sur la Transformation de Tschirnhausen,” and 
“ Deuxieme Note sur la Transformation de Tschirnhausen,” Grelle, t. lviii. pp. 259 and 
263 (1861), [273 and 274], obtained the transformed equations for the cubic and quartic 
equations; and by means of a grant from the Government Grant Fund, I have been 
enabled to procure the calculation, by Messrs Davis and Otter, under my superintend 
ence, of the transformed equation for the quintic equation. The several results are 
given in the present memoir; and for greater completeness, I reproduce the demonstra 
tion which I have given in the former of the above-mentioned two Notes, of the 
general property, that the function of у is an invariant. At the end of the memoir 
I consider the problem of the reduction of the general quintic equation to Mr Jerrard’s 
form x 5 + ax+ h = 0. 
Considering for simplicity the foregoing two equations 
(a, b, c, d, e\x, l) 4 = 0, 
у = {ax + b)B + {ax 2 + 4bx + 3c) G + {ax 2 + 4bx 2 + 6cx + 3d) В; 
let the second of these be represented by y=V, the transformed equation in у is 
(y-V 1 ){y-V 2 ){y-V a ){y-V 4 ) = 0 > 
where V ly V 2 , F 3 , F 4 are what F becomes upon substituting therein for x the roots 
x 1} x 2 , x 3 , Xi of the quartic equation respectively. Considering i/ as a constant, the 
conditions to be satisfied in order that the function in у may be an invariant are 
that this function shall be reduced to zero by each of the two operators 
ад^ + 299 c + ЗсЭй + 4d9 e — ( Dd c + 26 У 9 Д ), 
499 ft + Scd/j + 2d9 c + ed d — (2 Cd n + -B9 C ). 
These conditions will be satisfied if each of the factors y— V u &c. has the property 
in question; that is, if y—V, or (what is the same thing) if F, supposing that x 
denotes therein a root of the quartic equation, is reduced to zero by each of the two 
operators. Considering the first operator, which for shortness I represent by 
А-{Пд с + 2Сд л ), 
in order to obtain AV we require the value of Ax. To find it, operating with Д on 
the quartic equation, we have 
(a, b, c, d\x, 1) 3 Дж + (а, b, c, d\x, 1) 3 = 0,