Wilder’s Algebraic Solution. 271 
Art. XI.—Algebraic Solution ; by Mr. C. Wixper, of New 
Orleans. 
Remarks on the determination of y, in y"+-ay"—'! +-by"-? 
+cy"—3+dy"—* ... +ky+l=0. 
(A) 
By’ and then deter- 
mine S,, so that (B) may be a Sebi of (A), independently 
of x, we shall have 
z?+8, 
If we assume the Gnedonc Po age 
2-9? 
wey 
by writing y+-z for ye it is changed to 
ek 5 
=e 
w+y+z 
which gives to (A) the form of eRe eae 
*+ay 
Now if we make (B)=0, we ‘atiall-o mn have Ea) sabe hence, 
y comparison, 
» (1), 
z4 epee (2), these two equations 
together with rt+y+2=0, (8), are sufficient to deter- 
mine y; for from hs pag d (2), 
a? - 
and from (1) and (B), y= =5 a which is the co aaciaes 
In like manner, — the function 
at+8,7°+8, (C) 
x?+yz+p (Dy 
we shall have, when (D) i ue factor of ios oe of x, 
: — 3py)x* 
MR GE 
writing yte for y, and it is changed to 
x® +(y* +3zy ESS ey ee te 
v2 -+-Yyr+p 
Now when (D)=0, we also have (C)=0, and we may there- 
fore, after dividing se by x*, compare it with the equation 
ay? -+by-+-c= 
which gives 3z=a, (1) 
32? ek (2) 
