186 AN EIGHTH MEMOIR ON QUANTICS. [405 
324. There remains only the subimaginary transformation, viz. this has been reduced 
to x : y into kX : Y, the transformed equation is 
Y) n — 0, 
and this will be a real equation if some power k p of k (p not greater than n) be 
real, and if the equation (a, y) n = 0 contain only terms wherein the index of x 
{or that of y) is a multiple of p Assuming that it is the index of y which is a 
multiple, the form of the equation is in fact x a (x p , y p ) m = 0, (n = mp + a), and the 
transformed equation is X a (k p X p , Y p ) m = 0, which is a real equation. 
325. It is to be observed that if p be odd, then writing k p =K (K real) and 
taking k' the real p-th root of K, then the very same transformed equation would 
be obtained by the real transformation x : y into k'X : F; so that the equation 
obtained by the imaginary transformation, being also obtainable by a real transfor 
mation, has the same character as the original equation. 
326. Similarly if p be even, if K be real and positive, the equation k p = K has 
a real root k' which may be substituted for the imaginary k, and the transformed 
equation will have the same character as the original equation ; but if K be negative, 
say K= — 1 (as may be assumed without loss of generality), then there is no real 
transformation equivalent to the imaginary transformation, and the equation given by 
the imaginary transformation has not of necessity the same character as the original 
equation; and there are in fact cases in which the character is altered. Thus if p = 2, 
and the original equation be x (x 2 , y 2 ) m = 0, or (x 2 , y 2 ) m = 0, then making the transfor 
mation x : y into iX : F, the transformed equation will be X (.X 2 , — F 2 ) m = 0 or 
(X 2 , — F 2 ) m = 0, giving imaginary roots X 2 + ciY 2 = 0 corresponding to real roots x 2 — ay 2 = 0. 
Article No. 327.—Application to the auxiliars of a quintic. 
327. Applying what precedes to a quintic equation (a,. . . .fpx, y) 8 =0, this after 
any real transformation whatever will assume the form (a,. . .$V, yf = 0; and the only 
cases in which we can have an imaginary transformation producing a real equation of 
an altered character is when this equation is (a, 0, c, 0, e, 0$V, y') 5 = 0 (c‘' not = 0), or 
when it is (a, 0, 0, 0, e', 0$V, y'f = 0, viz. when it is x'(ax 4 -\-10c'x 2 y' 2 + oe'y 4 ) = 0, or 
x'(a'x >4 + 5e'y' 4 )=0. In the latter case the transformation x, y' into X "V — 1 : F gives the real 
equation X (aX 4 — 5e'F 4 ) = 0. I observe however that for the form (a', 0, 0, 0, e\ 0$#, y) 4 , 
and consequently for the form (a, ... $#, y) 6 from which it is derived we have 
J = 0; this case is therefore excluded from consideration. The remaining case is 
(a', 0, c, 0, e', 0$V, y') 5 = 0, which is by the imaginary transformation x' : y into iX : F 
converted into (a, 0, -c', 0, e', 0$X, F) 5 = 0 ; for the first of the two forms we have 
X = 16 a'ce-' 2 , and for the second of the two forms J = - 16 a'c'e' 2 , that is, the two values 
of J have opposite signs. Hence considering an equation (a, b, c, cl, e, f^x, y) s = 0 for 
which J is not = 0, whenever this is by an imaginary transformation converted into 
a real equation, the sign of J is reversed ; and it follows that, given the values of 
the absolute invariants and the value of J (or what is sufficient, the sign of J), the