245 
we have 
F,, (A) Men= 1 
nz y = > ——___.  —___. 0; 
that (A) en” Ub, F(@-1) + 7 ow 
or, if a1, a2, a3, &c., be, as before, the roots of F, (p)=0, 
Mer 
: + C, (a, +1)7 + OC; (a, +1)*+ &e. 
F(a ) ene = 
+H (ay? * W(e@=1) 
Hence, at once, 
Me. F, F(A) on F,, (A) 2 tn (A) nx mx 
ex caeayl ae F(A) “ke “F(a)’ ~ &e, om, 
{fel 
Go F, (A) ent F(A) e of, (A) on en - &e. (a+1)?, 
zo(1- “Ray” + F(a)” FA 
or, finally, 
Sain ine ea | 
& Fea! Fea) 
we (em —1) é F(e" — 1) ane tos 
| + (em? —1). B (e™"-1) 
4 Ee 
n (atl — 
ZC 1)" hee Hise gee 
(at ) F, (ea+1-1) 
ne (atl = 1) u F, (e" a+1—1) err _ &e, 
Fi (2" a+1 = 1) . Ff, (e?a+1-1) 
EXAMPLE. 
Let it be proposed to solve the equation in finite differences, corre- 
soe eng to Pfaff’s differential equation 
(a+ a'p*) Au, + (b+ bp”) Aue + (6 + ep) Up = X= ZM'e"™ 
This equation may obviously be thrown into the form 
(A - 4) (A- a) u, + (A— Py) (A - B2) pu, = =Me 
Hence we obtain the primary form— 
4 SSE By) (A- 2) a a ae Piss 
1+k (A-a,) (A- Ee jue TS | CST Be ) 
+C,(1+ a2)" 
