562 
lilE. A. CAYLEY ON TSCHIENHAITSEN’S TEANSEOEMATION. 
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 Ottee, under my superintendence, of the transformed equation for the 
quintic equation. The several results are given in the present memoir ; and for greater 
completeness, I reproduce the demonstration which I have given in the former of the 
above-mentioned two Notes, of the general property, that the function of y is an inva- 
riant. At the end of the memoir I consider the problem of the reduction of the general 
quintic equation to Mr. Jerraed’s form Cif-\-ax-\-h=-Q. 
Considering for simplicity the foregoing two equations 
{a, h, c, d, ejx, 1 )^ — 0 , 
{ax-\-h)Q -h [ax‘^-\-^hx-\- ^c)C-\-{ax'^-\-^hx'^-\-Qcx-\- 3<?)D ; 
let the second of these be represented by y=V, the transformed equation in y is 
(y-V:)(y-V,)(y-V3)(2/-V,)=0, 
where V,, Vg, V 3 , V 4 are what V becomes upon substituting therein for x the roots 
xl, X 2 , X 3 , Xi of the quartic equation respectively. Considering y as a constant, the con- 
ditions to be satisfied in order that the function in y may be an invariant are that this 
function shall be reduced to zero by each of the two operators 
— (D^c “l"2CdB), 
— (2CdD-j-BBc) • 
These conditions will be satisfied if each of the factors y—^i, &c. has the property in 
question ; that is, if y—Y, or (what is the same thing) if V, 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 
A-(DBc+ 2 CdB), 
in order to obtain AV we require the value of Ax. To find it, operating with A on 
the quartic equation, we have 
(«, b, c, dy^x, VfAx-\-{a, b, c, d^x, 1)^=0, 
or Ax= — 1. In AY, the part which depends on the variation of Ax then is 
— «B fi- (— 2(za’— 4J)C-f ( — ^ax^— 8 ^. 2 ;— 6 c)D, 
and the other part of AV is at once found to be 
-\-dQ-\-{iax-\-Qb)C-{-{'iao^-{-Vlbx-\-^G)D ; 
whence, adding, 
AV=2(«.T+J)C+(fa'^+45a’-f3c)D, 
and this is precisely equal to 
(DBe-l- 2 CBB)V; 
so that V is reduced to zero by the operator A — (DBc+2CBb)‘ 
