176 
AN EIGHTH MEMOIR ON QUANTICS. 
[405 
line x — \y = 0 lies between the lines x + y = 0, ¿c —2y = 0, and so does not cut either 
the region P or the region Q) restricts (x, y) to the region P ; and for every point 
of P y is at most = 1, and the condition J-§ Â — y = + is of course satisfied. The con 
dition, 125 J 2 — 9P = +, is thus wholly unnecessary, and omitting it, the conditions are 
P = +, J = —, 2 n P — J 3 — f JD = 0, character 5r, 
which, — | being an admissible value of ¡i, agrees with the result ante, No. 283. 
307. It may be remarked in passing that if 12345 is a function of the roots 
(a?!, x 2 , Xo, x 4 , x 5 ) of a quintic equation, which function is such that it remains 
unaltered by the cyclical permutation 12345 into 23451, and also by the reversal 
(12345 into 15432) of the order of the 
12 values 
a, = 12345, 
«„= 13425, 
a 3 = 14235, 
a 4 = 21435, 
a 5 = 31245, 
= 41325, 
roots, so that the function has in fact the 
ft = 24135, 
ft = 32145, 
ft = 43125, 
ft = 13245, 
ft = 14325, 
ft = 12435, 
then <f) (a, ¡3) being any unsymmetrical function of (a, /3), the equation having for its 
roots the six values of </>(a, ¡3) (viz. (sq, ft), <£(a„, ft)...$(a s , ft)) can be expressed 
rationally in terms of the coefficients of the given quintic equation and of the square 
root of the discriminant of this equation. In fact, v being arbitrary, write 
L = H 6 {v — (f) (a, /3)}, M=U e {v — (f>(@, a)}, 
then the interchange of any two roots of the quintic produces merely an interchange 
of the quantities L, M; that is, 
L + M and (L — M) -r £4 (#i, x. 2> x 3 , x t , x 5 ) 
are each of them unaltered by the interchange of any two roots, and are consequently 
expressible as rational functions of the coefficients; or observing that £4 (x u x 2 , x 3 , x i} x 3 ) 
is a multiple of VP, we have L a function of the form P+Q'Jl); the equation 
P = 0, the roots whereof are v = <f> (a 1} ft) ... v = </> (a 6 , ft), is consequently an equation 
of the form P + QftP = 0, viz. it is a sextic equation (*^v, 1) 6 = 0, the coefficients of 
which are functions of the form in question. Hence in particular 
it 2 = 12345 = (x x — x 2 f (x, — x 3 Y (x 3 — x 4 ) 2 (x 4 — x 3 ) 2 (x 5 — x 4 f 
is determined as above by an equation (*j£ti 2 , 1) 6 = 0. Another instance of such an 
equation is given by my memoir “ On a New Auxiliary Equation in the Theory of 
Equations of the Fifth Order,” Phil. Trans, vol. CLI. (1861), pp. 263—276, [268].