618 
PEOEESSOE STLYESTEE ON THE EEAL 
So when the canonizant has two equal roots and is of the form C {x-\-]py){x-\-gyY ; in 
which case the reduced form is au 5 -\-heuv 4 The canonizant in respect to u, v 
becomes 
a 0 0 0 
0 0 0 e 
0 0 e f 
v 3 —vhi vu 2 —u 3 . 
i. e. ae 3 uv \ Hence, writing 
u=.x-\-jpy , v=x-\-qy, F=au 3 -\-heuv 4 +fv 5 , 
a , e, f may be obtained, as before, by means of three linear equations, and the terms 
au 5 , 5euv 4 , fv* form a single and unique system. 
Finally, when the canonizant vanishes entirely, so that the form becomes au 5 -\-fv 5 , 
the quadratic covariant will take the form C(x-\-ey)(x-\-fy) ; and making u=x-\-jyy, 
v=x-\-yy , a,f become determined by means of two linear equations, so that an 5 , fv 5 
form a single and unique system, as in the preceding cases. 
(37) When the canonizant has three distinct roots, they may be all real, or one real and 
the other two imaginary. In the former case, in the expression nf-\-sv 3 -\-tw 5 , u, v, w may 
be considered as all real functions of x, y , and r, s, t will then also all of them be real. 
In the latter case w may be taken as a real function of x, y, u, v as conjugate imaginary 
functions ; and consequently it is easy to see that, except when r, s are equal to each 
other, they will constitute a pair of conjugate imaginary quantities : in this case we may 
take for our canonizant form 
or, if we please, 
r(=^y+s(=^y+tu‘; 
rtf+sVi + tw 5 
understanding by u p v t — u - + w , — u w respectively. And it should be noticed that 
the determinant of u p v t in respect to u, v will be 
1 [ 
2 2 
_1 -i 
2 IT 
which is i. 
(38) Let us proceed briefly to express the invariants of ru 3 -\-sv 5 -\-tw 5 , which call O, 
with respect to m, v; the corresponding ones of ru 3 -^-sv 3 -\-tw 5 , which call <t> ; , in respect 
to the same variables u, v will be found by attaching to these suitable powers of i. 
0=(r— t, —t, —t, —t, —t, s — t'Jfcu, v)\ 
