PROFESSOR CAYLEY’S EIGHTH MEMOIR ON QUANTICS. 
549 
323. For the superimaginary transformation, starting in like manner from 
(«,... £.r, y)"=0, this may be expressed in the form 
(a+fli, y+eii, . . . , y—tii, u—fii$x-\-iy. x—iy) n = 0, 
viz. when in a real equation (x, y ) n — 0 we make the transformation x : y into x-\-iy : x—iy , 
the coefficients of the transformed equation will form as above pairs of conjugate ima- 
ginaries. Proceeding in the last-mentioned equation to make the transformation 
x+iy : x—iy into #(X-f ^Y) : X— iY, I throw k into the form 
cos 2 <p-H sin 2<p, =(cos p-\-i sin <p) -r-(cos Q — i sin <p) 
(of course it is not here assumed that <p is real), or represent the transformation as that 
of x-\-iy : x— iy into (cos sin <p)(X~HY) : (cos <p — i sin <p)(X-> Y”) ; the transformed 
equation thus is 
(ct-\~pi, ... a— (3i£(cos(p-\-i sin <p)(X-f <Y), (cos p— i sin <p)(X— «Y))"=0. 
The left-hand side consists of terms such as (X 2 -f-Y 2 ) n_2s into 
(y -|-A*) (cos sp -J- i sin s<p)(X -J- i Y) s + ( y — c^)( co s — i sin sp) (X — iY ) s , 
viz. the expression last written down is 
=(y cos s<p— l sin s<p){(X+«Y) s +(X— *Y) S } 
— (y sm cossip)<- 
and observing that the expressions in { } are real, the transformed equation is only real 
if (y coss<p — &sins<p)- 7 -(ysins<p+&coss<p) be real, that is, in order that the transformed 
equation may be real, we must have tans<p=real ; and observing that if tan s<p be equal 
to any given real quantity whatever, then the values of tan <p are all of them real, and 
that tan <p real gives cos p and sin p each of them real, and therefore also p real, it 
appears that the transformed equation is only real for the transformation 
x-\-iy : x— m/=(cos p-\-i sin p)(X.-\-iY) : (cos p— i sin <p)(X— *Y), 
wherein p is real ; and this is nothing else than the real transformation x : y into 
X cos p — Y sin p : X sin <p+Y cos p. Hence neither in the case of the neutral trans- 
formation or in that of the superimaginary transformation can we have an imaginary 
transformation leading to a real equation. 
324. There remains only the subimaginary transformation, viz. this has been reduced 
to x : y into kX : Y, the transformed equation is 
(«,... X*X, Y)»=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 , . .yjc, y) n = 0 contain only terms wherein the index of x (or that 
of y) is a multiple oip. 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. 
