405] 
AN EIGHTH MEMOIR ON QUANTICS. 
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822. lhe original equation after any real transformation thereof, is still an equation 
of the form 
(a, y) n = 0; 
and it we consider first the neutral transformation, the transformed equation is 
(a, ...$X + kY, Y) n = 0 ; 
this is not a real equation except in the case where k is real. 
323. F or the superimaginary transformation, starting in like manner from 
{a, ... \x, y) n = 0, this may be expressed in the form 
(a + f3i, y + Si,... , 7 — Si, a — ftifyx + 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 imaginaries. Proceeding in the last-mentioned equation to make the trans 
formation x + iy : x — iy into k(X + iY) : X — iY, I throw k into the form 
cos 2<p + i sin 2(f), = (cos (f> + i sin cf>) (cos (f> — i sin (f>) 
(of course it is not here assumed that </> is real), or represent the transformation as 
that of x + iy : x — iy into (cos cf>+ i sin <f)) (X+ iY) : (cos cf) — i sin cf>)(X — iY) ; the trans 
formed equation thus is 
(a + (3i, ... a — /3f][(cos (f) + isin <£)(X + iY), (cos cf>-i sin <£)(X — iY)) n = 0. 
The left-hand side consists of terms such as (X 2 + F 2 ) 7 » -2 * into 
(7 + Si) (cos scf) + i sin s<f>) (X + iYy + (7 — Si) (cos scf> — i sin scf>') (X — i Y) s , 
viz. the expression last written down is 
= (7 cos S(f> — S sin scf)) {(X + iYy + (X — iF) s } 
and observing that the expressions in ( j are real, the transformed equation is only 
real if (7 cos scf> — S sin S(f>) -h (7 sin scf) + S cos .9</>) be real, that is, in order that the trans 
formed equation may be real, we must have tan S(f) = real; and observing that if tan scf> 
be equal to any given real quantity whatever, then the values of tan (f> are all of them 
real, and that tan cf> real gives cos cf> and sin (f> each of them real, and therefore also cf) 
real, it appears that the transformed equation is only real for the transformation 
x + iy : x — iy = (cos (f> + i sin </>) (X + iY) : (cos (f) — i sin </>) (X — iY), 
wherein (f> is real; and this is nothing else than the real transformation x : y into 
X cos </> — Y sin (f> : X s\n <f> + Y cos cf). 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. 
C. VI. 
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