550 
PROFESSOR CAYLEY’S EIGHTH MEMOIR ON QTJ ANTICS. 
325. It is to be observed that \f p be odd, then writing k v — K (K real) and taking k' 
the real j?-th root of K, then the very same transformed equation would be obtained by 
the real transformation x : y into ^X : Y ; so that the equation obtained by the imaginary 
transformation, being also obtainable by a real transformation, 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?,y 2 ) m = 0, or (x 2 ,y 2 ) m = 0, then making the transformation x : y into iX : Y, the 
transformed equation will be X(X 2 , — Y 2 ) m =0 or (X 2 , — Y 2 ) m =0, giving imaginary roots 
X 2 -j-aY 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 («, .... ~fx, y) 5 =0, this after any 
real transformation whatever will assume the form (a 1 , . . . fx\ y'f= 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 1 , 0, c', 0, e', Ofx', y'y=0(c' not = 0), or 
when it is ( a 0 , 0, 0, e', Ofjtf, y') 5 =0, viz. when it is x'(a'x' 4 -{- 10c'x ,2 y l2 -\-5e'y 4 )=0, or 
a ; («V 4 + 5ey 4 ) = 0. In the latter case the transformation x', 3/ into X %/ — 1 : Y givesthe real 
equation X(a'X 4 — 5e'Y 4 )=0. I observe however that for the form ( a ', 0, 0, 0, e', Ofx, y ) 4 , 
and consequently for the form (a, .. . fx, y) b 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\ y) 5 =0, which is by the imaginary transformation x' : y' into iX : Y 
converted into (a', 0, — d, 0, e\ 0J[X, Y) 5 = 0; for the first of the two forms we have 
J=1 Qa'c'e 12 , and for the second of the two forms J = — 1 Qalc'e' 2 , that is, the two values of J 
have opposite signs. Hence considering an equation ( a , b, c, d, e,ffx, y ) 5 = 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 different real equations which 
correspond to these data must be derivable one from another by real transformations, 
and must consequently have a determinate character ; that is, the Absolute Invariants, 
and J, constitute a system of auxiliars. 
