26 
Proceedings of the Royal .Society of Edinburgh. 
[Sess. 
From 
x = 
C E 1 F 1 A 2 CEB F/B\ 2 A/CV 
B B x B a 2 B' B + B C ~ B\C/ B\B/ 
we get the operators D^ 1 . D^ 1 • Pcb FT 1 • Eb 2 • Deb Pa 1 • Pc 1 * E B 2 > so that this 
solution will also exhibit different denominators and zero elements. 
Finally, from 
B C1_E1_F1_ == _B C y A_E/A\ 2 F/A\ 3 
A - Ax Ax 2 Ax 3 ~ A + A B A\B/ + A\B/ 
we find as operators 1 . Dp 1 . D B , • El 2 • D B > and E^ 1 . Dp 3 . D B , a quartic, 
cubic, and quadratic operator, and the solution proceeds in descending 
powers of one coefficient only, viz. B. 
The coefficients occurring in a development by a quadratic or cubic 
operator we have already learned, viz. 
E 
C 
BE 2 0 B 2 E 3 , ,B 3 E 4 . { B 4 E 5 , 0 B 5 E 6 , 1QO B 6 E 7 
+ 2 « 1+ 5-—^- + 14—7— + 42—— + 132- 
C 3 
c 5 
C 7 
C 9 
C 11 
C 13 
etc., 
for the quadratic, and 
E AE 3 0 A 2 E 5 
C C 4 
10 A 3 E 7 _A 4 E 9 0 „ Q A 5 E 41 , , )oq A 6 E 13 , 
C 7 - 12 ci ( r + 55 era - 2/3 -^r+ 1428 -c^- etc -> 
for the cubic. (A cubic operator, with its three differentiations, is intrinsi- 
cally — etc.) Those belonging to a quartic operator are 
I * + *<%+»!%■+ •«+ * ■ ™ 84 "“, *. 
The direct solution of the quartic may be represented — on the flat at 
least — as follows : — 
From the operators 
IV, -IV- IV, IV .IV 2 . IV, DiMV.DF on 
we have 
x = 
F 
E 
CF 2 BF 3 
E 3 + E 4 
C 2 F 3 CBF 4 B 2 F 5 
t >— —+5-^-3 
E 5 E 6 
etc., etc. 
E 7 
6 
AF 4 
E 5 
CAF 5 BAF 6 
E 7 
+ 7 
E 8 
C 2 AF 6 „ CBAF 7 B 2 AF 8 
28 ^9 +/ 2 jgio -45 j, n 
A 2 F 7 
E 9 
CA 2 F 8 BA 2 F 9 
4^ gii + 55 g 12 
„ C 2 A 2 F 9 CBA 2 F 10 
- 275 — -r,! 3 +858- 
E 13 
etc., etc. 
E 14 
etc., etc. 
