244 
Mum (m +” — Bi) (m+ n— Be) 
| = (m - ay) (nm — fue a1) (m+ nae 
(m + 2n — Bi) (m + 2m — Bz) (m +n — Bi) (m+ 1 — Be) 
zn — & | 
ga (m + 2n — an) (m+ 2n — ay) (m+n — a) (m+ n—ar\ 
+ 
(a1 + 2 — Bi) (m+ 2— Be) 
ay = EE oS rs 
“= ree {1 I n (” + a1 — az) 
LB (a; + 2n — Be) (ai + 2” — Bo) (1+ ”— Bi) (a1+n—- B2) on gis 
Qn? (2n + ai — az) (2+ a1 — ag) 
+ 
z (az + ” — Bi) (a2 +2 — B2) 
Aye 1 - n (n + ag — a) 
2 (a2 + 2n — Br) (a? + 2n — Be) (a2 +” — Bi) (az +n” — Bz) oe 
+ 2n2 (2n + a2 — a1) (N+ az — a1) 
lo 
n — &e. 
It may be remarked that the same method of solution, precisely, applies 
to partial differential equations of the type 
F, (xD, + yD, + 8D, + &.) u+ F, @D, + yD, + &e.) Ou = 2On 
where 9,, ©,, are given homogeneous functions in the independent va- 
riables of the degrees n, m, respectively ; or to the reducible type 
F, (axD, + byD, + &e.) uw + ¥F,, (axD, + byD, + &e.) ©,u = 2p, - 
and that the partial differential equation, corresponding to the total 
differential equation discussed by Pfaff, namely 
(a +a/O,) (2? D?, + 2xyD,D, + y?D*,) u + (6 + V'O,) (xD, + yD,) u 
+ (¢+ O,) U=2Om, 
where ©, is a given homogeneous function in x and y, admits of easy 
solution by a process similar to that already exhibited. 
4, A similar method of solution will apply to the class of equations 
in Finite Differences represented by the typical form: 
F, (A) u, + F,,(A) e* u,= =e", 
where, as before, F’, and F,, represent algebraic functions of the symbol 
A, and the degree of A in F, is supposed to be at least not higher than 
in F,. In fact, since 
f(A) Car = f (a—-1) Ca", 
