The collected mathematical papers of Arthur Cayley.

405]

AN EIGHTH MEMOIR ON QUANTICS.

165

But if D=+, J=-, then,u being any number at pleasure between the limits + 1
and - 2, both inclusive, if
21L - J3 + PJD = +, the character is 5r,
21L - J3 + CtJD =-,,, r + 4i.
284. The characters 5r of the region P and r+4i of the regions Q and T may
be ascertained by means of the equation (a, 0, c, O, e, 00, 1)5 = 0, that is
0(ac + 10c2+o5e) =0;
there is always the real root 0 = 0, and the equation will thus have the character 5r
or r+ 4i according as the reduced equation a04 + 10c02 + 5e = 0 has the character 4r
or 4i. It is clear that (a, e) must have the same sign, for otherwise 02 would have
two real values, one positive, the other negative, and the character would be 2r + 2i.
And (a, e) having the same sign, then the character will be 4r, if 02 has two real
positive values, that is, if ae - 5c2 is =-, and the sign of c be opposite to that of
a and e, or, what is the same thing, if ce be =-; but if these two conditions are
not satisfied, then the values of 02 will be imaginary, or else real and negative, and
in either case the character will be 4r.
285. Now, for the equation in question, putting in the Tables b=d=f=O, we find
D =256 ae3 (ae - 5c2)2,
J= 16 ce(ae + 3c2),
212L - J3 = 212 e3 {2 (ae- c2)4 - c2 (ae + 3c3)3}
= 212 ce3 (ae - 5c) (2a3e3 ac2e + 8aa4e + c6).
We have by supposition D=+, that is, ae=+; hence J has the same sign as ce;
whence if J=+, then also ce=+, and the character is 4i; that is the character of
the region T is r+4i. But if J=-, then also ce=-. But ae being =+, the sign
of 211L-J3 is the same as that of ce(ae - 5c2), and therefore the opposite of that of
ae - 5c2: hence D = +, J = -, the quartic equation has the character 4r or 4i
according as 211L-J3 is =+ or =-. Hence the region P has the character 5r and
the region Q the character r + 4i; and the demonstration is thus completed.
Article Nos. 286 to 293.-HERMITE'S new form of TSCHIRNHAUSEN'S transfor2mation, and
application thereof to the quintic.
286. MI. Hermite demonstrates the general theorem, that if f(x, y) be a given
quantic of the n-th order, and (x, y) any covariant thereof of the order n - 2, then
considering the equation f(x, 1) 0, and writing
_ (x, 1)
fj(x, 1)