of the Fifth Degree. 199 
But the preferable course will be to proceed by elimination as in 
the present ™ discussion. 
83. Let yw! and y be the values which uv and v take when P—S 
is substituted for$, The indicated form for the root of a general 
quintic is 
M+Hi" Vn+ Vv+i” lech v¥ 
+i" Tu— Vv +t" Vyl— viii 
84, This expression coincides in substance with that differ- 
ently deduced by Mr. Jerrard in his ‘Essay.’ It embraces the 
second solvable form of Euler (Novi Comm. Petr. p. 96 et seq.), 
as it probably may be made to do that of Abel (posthumous 
theorems, Crelle, vol. v. p. 336). It embraces also the first 
soluble form of Euler as well as that of Demoivre, and the one 
that I have calculated by making one only of Lagrange’s func- 
tions vanish (Diary for 1858). 
85. The vanishing of this function is marked by 3=0, and 
the roots of the form last mentioned are comprised in the ex- 
pression} 
@ AU. feeaenme kee 
ro es es a es 
oF sages a P ax ta/ (2 PA ber). — Pps 
4 
op. om PA FEF ax) — Ps, 
2 
86. Presumably $, is a rational function of $,, and indeed I 
* The mode of elimination that I have found to be the most convenient 
in practice is Newton’s, in which we annihilate extreme terms alternately. 
The process used in the text is a modification of Newton’s, arrived at thus: 
Let X and Y be of the mth and mth degrees in the quantity to be elimi- 
nated, and let X, and X2 be indeterminate expressions of the (n—m)th 
degree in the same quantity. Form the expressions 
X,X+Y and X,X+Y, 
and assign the indeterminate coefficients so that the first n—m-+-1 terms of 
the former and the last n— aaa of the latter shall vanish. If we make 
—a=f, #4+Qa=g, 
the unmodified method A Newton gives the cubics 
Sahb§—4H( fa? + 29° ab? — 24 gb +H7( fy —43a5)a?=0, 
(fg—45'0°)a'b? + 23fa"b?—435(93 + 2a") a3b—g?°=0 
in place of those in the text. 
t A and B (which I call respectively v and w in the ‘ Diary’) are known 
and rational, but complicated functions of the coefficients. Compare art. 21 
of Mr. bail s paper on Symmetric Products in the Manchester Memoirs, 
vol. xv. 
P2 
