‘Wilder’s Algebraic Solution. 273 
b ee eee ; eee 
%° ort ++ (2 —7)? —Gqg=9, which reduced is given in 
most books of algebra. 
o24+S, a x3,, (G) 
a per” (Hy which 
becomes, = (H) is a factor of (G) independently of x, 
ea ly*—4py? +4qy+2p?)zx® 
LF -+yx? +prt+q 
(p* — Aud *p+2q7y?)v*—q*, 
Again, let us assume 
oo 
+pxr+q 
(H1)=0 and of course (@)= 0, we have by comparison with 
y* +by? rey +d=0, after having divided 
(G) by x? spt mb, (1) 
In? 
va: de a=, (2) 
4G 
992 2 Ps a a2) sted, (3): 
ee x-* from (1 sa (2) by the process indicated 
2p? (1) +4 (2), we have sp?+4q?+2bp? —cq=0. This 
eqnslion is satisfied by making 4p+-b=0, (4), and 4g—c=0, 
e three equations, (3), ) and (3) pas to (H), are 
sufficient to determine eB os 3) w 
x'24(d—2p?)x*+(p*+ 15tyeegs ea then we obtain by 
the second example, x* =elbed), and from si 
ee oe Gite Sine 22 
we orideatly wey Agqyp? we y*sa*; identically nothing, 
otherwise, there would be a relation between a, 6 and c, 
which is not the case. 
G) 
Let us write in oe Qy for y, and 2y2—p for p, which 
changes this function to 
212-+-8y4—8py? “8qy-_2p2)x*+(16y° —32py °-32qy> +24 p?y* 
x 24 -+-2yxz? 
+32pqy* + (169? — 8p*)y? — Sp*qy +p* +4pq? )e* —@" 
+(2y*—p)a+q 
and then we have, by a comparison with the function - 
y® tay® +by® +cy! +dy*+ey? tfy+e=0, 
Vor. XVI—No. 2. 
