Wilder’s Algebraic Solution. 281 
it would be useless to repeat the calculation by which we 
have determined the functions, 8,,8,,,5,,3 it 1s sufficient 
to say that (A’) will be, 
ps 
—sqyp? 
+y° ] +5ry*p? +p: | 
— 5py* — Srp* — d5prq* 
+5qy? § +5q?p? +5r?4q? 
x*°+4 5py pre?  v19 — Sp? r2gq rip ; 
— dry — d5qry* —5r*q 
— Spq —5q*y —d5pr°y 
+5p7y? 
—10r?p 
+5q?r J 
and B’, 24+-+-yx*+pa0?+qa+r. 
Let us now suppose that such a function o(apgr) is 
written for y, as renders «2 °(y* — 5py? +5qy? + 5p?y — dry 
— 5pq)x'* +-(ps — Spe oe p? —10r2p+-5q?r)x'°+(q> — 
5prq* +op* rq —5r°q)c*+r>=0, we may then compare it 
with y* +by* +-cy? +dy+e=0, after dividing the first equa- 
tion by «5; Hero done we have, 
=—b, 1 
ae we (2), 
= (3); 
ais 7. ms § at spe kg? E = 10rtp = set 
(q oprapt—50'0) 5 a 4), 
or bat 
x? ile, spayeteG 78 bp th5q? ‘p? —10r?p — 5q?r)x'°+ 
(q° — 5prq?-+5p*r2q —5r*q)z*+q° =0. 
These four equations, joined to sph Set EE 
are sufficient to determine y. 
It is very evident, pat it is a matter of indifference, which 
ofthe letters y, p, q, or r, we treat as the pBkngwe+ for they 
re all of five di naire in the function before u 
This remark is applicable to other cases ; as, ea example, 
ia 8 Br 2 aac 2 ; here, ma- 
_2?+yr+p : 
5 
the function (D)’ which is 
Vor. XVI.—No. 2. 
