Wilder’s Algebraic Solution. Q75 
making the absolute term equal to W &.y* —py? —qy +p? 
4 
4-V 4pq?x' — x"? mi Bee —8qy — 2p*)x*+q* _, 
reducing «1*-++-(8y* — Spy? = Sae > 2p? je? (pay hig 
*p)x*—g*?=0, but By! —S8py* —8qy=8V 2; hence, 
our re i 
#14 — (ap? —8V g)2"4(p?—4V g —4g2p)xt —q' = 
which is the product of the factors no and (6)= 0, re- 
sulting from the transposition of —4 
rom the foregoing examples, one ould be led to think 
the method pursued here was applicable to all rational al- 
gebraic equations; but let us, before we attempt to follow 
the analogy, recall, and demonstrate the following proposi- 
tions. 
HMO VAS, 2M IAS | MO DLS. em™n—a) 
L) Lyg2 +pxs +qx"~* 
First, let 
» FSco—2 yt” + Seni ym_ a 
- pir+-u 
be a function in which 2. DyY; Ss etc. are independent functions 
of any number of other quantities whatever, then I sa 
Sm, Sams Sam etc. can tly al es (B) shall functions o of 
etc. independently of x, so that a a 
OHA: for for, continue the operation indicated till the index of 
zx in the remainder is n—2, and then make the poms 
equal to zero independently of Li which can always be done, 
since the whole number of unknowns, Sm, Som, Sam, ete., 
and the whole number of éonatiGas ism—1; and itis ‘evident 
that they are of the first degree, relative to Sins Se amsote. 
It is plain that the function (A’) may be decomposed in 
(n—1) factors, ea ) Pp ee ("+7") (2" +0") etc. ; rat 
+h +7 
if we put Fey ese 
G4 ay <p 
007+ 088-+ Byi-apd etc. =q 
aByd etc. 
then adopting the notation a formula of Lacroix Comple- 
ments des Elemens d’Algebre, we have 
