PEOFESSOE CAYLEY’S EIGHTH MEMOIE ON QTJ ANTICS. 
547 
It is in fact by means of these comparatively simple canonical expressions that M. Hermite 
was enabled to effect the calculation of the coefficient 91. 
Article Nos. 318 to 326. — Theory of the imaginary linear transformations which lead to 
a real eguation. 
318. An equation ( a , b, c, . . .fx, y) n =- 0 is real if the ratios a:b:c, See. of the coeffi- 
cients 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, . . .fx, 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 transformation, 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-\-bi)x +(c + di)y : (e +fi)x + (g+hi)y, 
or, what is the same thing, I write 
x=(a-\-bi )X-b(c-)- di) Y, 
y=l e +/*') x + (9 + /w ') Y > 
and I seek to find P, Q real quantities such that Pa‘-f-Q^ may be transformed into a 
linear function BX+SY, wherein the ratio B : S is real, or, what is the same thing, such 
that BX+SY may be the product of an imaginary constant into a real linear function 
of (X, Y). This will be the case if 
Fx+Qy =(l+fl»){P(aX+cY)+q«X+^Y)}, 
that is if, 
P(&X+<ZY)+Q(/X+AY)= 0 {P(aX+cY)+QO»X+/Y)}» 
4 e 2 
