405] 
AN EIGHTH MEMOIR ON QUANTICS. 
183 
^ * act ^ means of these comparatively simple canonical expressions that 
M. Heimite was enabled to eftect the calculation of the coefficient ?(. 
a real equation. 
318. An equation (a, b, c,...'§x, y) n = 0 is real if the ratios a : b : c, &c. of the 
coefficients are all real. In speaking of a given real equation there is no loss of 
generality in assuming that the coefficients (a, b, c,...) are all real; but if an equation 
presents itself in the form (a, b, c,...Qx, y) n = 0 with imaginary coefficients, it is to be 
borne in mind that the equation may still be real; viz. the coefficients may contain 
an imaginary common factor in such wise that throwing this out we obtain an 
equation with real coefficients. 
In what follows I use the term transformation to signify a linear transformation, 
and speak of equations connected by a linear transformation as derivable from each 
other. An imaginary transformation will in general convert a real into an imaginary 
equation ; and if the proposition were true universally,—viz. if it were true that the 
transformed equation was always imaginary—it would follow that a real equation derivable 
from a given real equation could then be derivable from it only by a real transfor 
mation, and that the two equations would have the same character. But any two 
equations having the same absolute invariants are derivable from each other, the two 
real equations would therefore be derivable from each other by a real transformation, 
and would thus have the same character; that is, all the equations (if any) belonging 
to a given system of values of the absolute invariants would have a determinate 
character, and the absolute invariants would form a system of auxiliars. 
But it is not true that the imaginary transformation leads always to an imaginary 
equation; to take the simplest case of exception, if the given real equation contains 
only even powers or only odd powers of x, then the imaginary transformation x : y 
into ix : y gives a real equation. And we are thus led to inquire in what cases an 
imaginary transformation gives a real equation. 
319. I consider the imaginary transformation x : y into 
(a 4- bi) x + (c + di) y : (e +fi) x + (g + hi) y, 
or, what is the same thing, I write 
x = (a + bi) X + (c + di) Y, 
y = (e + fi) A +{g + bi) Y, 
and I seek to find P, Q real quantities such that Px+Qy may be transformed into 
a linear function RX + SY, wherein the ratio R : S is real, or, what is the same 
thing, such that RX + SY may be the product of an imaginary constant into a real 
linear function of (X, Y). This will be the case if 
Px +Qy = ( 1 + 0i) {P (aX + cY) + Q (eX +gY)},