334 
PROFESSOR A. R. FORSYTH ON 
so that 
and therefore 
tlwj( y " ,¥ -'- J )+6F„=t} 
\ v,y,z ) J 
SVdw —A /p p p\^^ 
j / 1 mj - 1 - /i> x p ' 
^ 3/> 2 
• (i). 
Let Z, X and Y be the eliminants of‘ F,„, F„ and F ;j respectively with regard to x and y, 
y and 2 , 2 and x ; then as explained in § 3 we may write 
X=A W1 F ; M +A«F„+A yJ F \ p 
L = B W F ' m + B„ F*+ B y ,F v 
Z = C ,„F ,u + C„ F w +C yj F p . 
Each of the equations 2 = 0, x=0, y= 0, has mnp solutions; let those values 
(mnp in all) which make F M , F„ and F /; vanish simultaneously be called congruous. 
Write 
A = 
A* A„ A p 
B„ t B„ B^ 
c» a a 
so that for non-congruous values A is zero. 
Now whatever be the value of T it can be put into the form 
'LQ, y, z) 
y, z) 
where <t> and F are rational integral algebraical functions of x, y and z ; and this can be 
expressed as 
U 
f( x ) 
where U is an integral function of x, y and 2 and f(x) is a function of x only. For it is 
fi=mn 
F(.r, y v Zy) n <f >(x, y p , z M ) 
(x=inn 
n ®(sc, zj 
n = l 
where y M , 2 ^ are a pair of values of y and 2 which satisfy the equations 
F«=0, F /t =0. 
!■ 
