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PROFESSOR SYLVESTER ON THE REAL 
Thus, then, if a— 0 and *= 0, all the coefficients, or else all except one, viz. b or e , are 
zero; 
if a = 0 and ^= 0 , all the coefficients, or else only not e and i or only not h 
or only not i are zero ; 
so if i= 0 and c= 0 , all must be zero except b and a or e or a; 
if c =0 and d— 0 , only e and i or else a and b or else a and i will differ 
from zero. 
Hence, then, in any case there will be at least four equal roots, or else F is of the form 
ax 5J ritf. 
Thus, then, for the first time has been here rigorously demonstrated, free from all 
doubt and subject to no exceptions, the following important proposition: 
Every binary quantic function not containing three or more equal roots is reducible to 
one or the other, of the two following forms, 
w 5 +fl 5 -J-w 5 , or au 5 -\~5euv 4 -\-fv \ 
The former is the case when the discriminant of the canonizant is different from zero, 
the latter when it is equal to zero ; for it will be observed that, whether the canonizant 
has equal roots or totally disappears, its discriminant in both cases alike is zero. 
(34) It has been seen that when the quintic has three equal roots the canonizant becomes 
a perfect cube ; and it may not be out of place here to point out what the conditions 
(necessary and sufficient) are to ensure the quintic having four equal roots. These are 
all comprised in that of the quadratic covariant vanishing. To prove this, let n be a factor 
of F(,£, y), so that 
F(#, y)=<&(x, *?)=(«, 0, y, &, e, OX#, rif- 
Then, since the similar covariant quoad x, y must also vanish, we have 
as — 4/3§+y 2 =0, — 3j3s+2y&=0, — 4ys+3^ 2 =0. 
If s=0, then &=0, y— 0 by virtue of the two extreme equations, and O, and therefore 
F, contains four equal factors. If s is not zero, 
38 2 8 3 58 4 5s /§ \ 4 
7— 47 ’ P= 2 ?’ a =l6T3’ and O becomes -ix(-x+2}}) ; 
so that, as before, there are four equal factors. Conversely, it is obvious that if there 
are four equal factors u, so that <P=au 5 -\-5bu 4 v, the quadratic co variant of O disappears. 
(35) The quadratic covariant also it was which led me to perceive the transformation 
applied in the antecedent article. For when the first minors of 
abed 
c d e f 
