S26 
PROFESSOR A. R. FORSYTH OH 
and E is therefore of the degree n(m —1), i.e., of the same degree in X as A; it is 
A . 
obviously of the same degree in y ; hence — is merely an arithmetical constant, and 
we may write 
A=E, B=F. 
2. When we come to apply this method to the formation of the eliminant with 
regard to y and ? of three equations F 1? F 2 , F 3 in three variables x, y, z the result, 
though of similar form, viz.: 
A^F \ +A,, 2 F 2 + A. rj3 F 3 , 
can in general be obtained neither so directly nor without the help of the considera¬ 
tions in Salmon’s c Higher Algebra,’ §§ 92, sqq. If the three equations be each of the 
degree 2, the method will apply exactly as in the preceding paragraph and we obtain 
where 
X—A. rJ F x -j- A, i2 F j 2 -J- A, 3 F 3 
A^_ r — "F ’ 
but if the equations be not of this degree, then the following is our rule. Let 
Fi, F s , F 3 be of the degrees m, n, p respectively : then we form all possible equations, 
which the variables satisfy, of degree not higher than m-\-n-\-p—2 : thus we multiply 
Fiby 
.n+p—2 n.n+p—3 
/ > U > 
. , y n+ P~h, y n +p~% . . . , y n+ P~h*, y ,l+ P- 5 z z , . 
and so on ; and so we obtain 
\(n -\-p— 1 ){n-{-p)-\-\(p-\-m— \)(p-\-m) — 1) 
equations from which to eliminate 
^(m+ n -\-p — 1) (m +n -\-p) 
quantities. But these equations are not all independent, being connected by a 
number of identities of the form 
z r y s F v F 3 = z r y s Fo. F x 
(where r-^s'Sp— 2), of which there are \p(p — 1); there are \n[n — 1) of the form 
z’YFvFg =z r'ys'Y^Y l where r'-\-s'^n — 2 
and |m(m-1) 
F 2 .F 3 =Ay'F 3 .F 2 where r"2 , 
and thus we have the proper number of equations. To find the eliminant X we write 
down the coefficients (which are, of course, functions of x) in the -g(n-Fy)— l)(n +P) + • • • 
equations ; and reduce them to the form of a determinant by adding the coefficients in 
