FINITE DIFFERENCES AND LINEAR DIFFERENTIAL EQUATIONS. 
275 
cleared of fractions, and so cleared to consist of a set of terms Then for this 
term (/o(I>))“'(«”'’G) ; 
and we have, as before, 
where Bo=l, and the law of formation is 
/o(w +p + +/i(w +J0 + +fln +/? + m)B,„_2 + . . . = 0 ; 
and the whole solution is found by taking all the values of/>. This will undergo a 
slight alteration, as the incipient term will be of the form yj)(D)(£^”~’‘^®G), where r factors 
of (2.) vanish by reason of some of the constant coefficients a„, &c. being zero. 
In like manner it would be easy to represent the series if G contained log x and 
its powers ; but for most other forms it would be necessary to expand G in order 
to represent the series explicitly. The solution however is theoretically complete, 
since it consists solely in the performance of operations which are known explicit 
functions of D. 
14. Before proceeding further with the main subject, I shall illustrate this process 
by a few examples. 
Ex. 1. Let the equation be 
( 1 + h^x+c^x"^) ^ + («i + = G. 
Referring to (2.), we have 
/oD=D(D-l),/iD= 62 (B-l)(D- 2 )+Ui(D-l), 
/2(D) = C2(D-2)(D-3)+^(D-2)+ao; 
roots of/’of=0 are 0 and 1. 
First complementary series, 
(-^ 0 + A 2 .r 2 -f- . . + A,„x”* -f . .), 
where 
Ao=l,andm(m-l)A„+( 62 (^-l)(^- 2 )+ai(w-l))A,„_, + (c2(m-2)(w-3)+&i(m-2)+ao)A^ 
is the law of formation. 
Second complementary series, 
where 
Aq = 1 , and (m + 1 ) w A,„ + (&2(m)(m — 1 ) + + (c2(m — 1 )( 7 w — 2) + ^>i(m — 1 ) + ^o) A„_2 = 0 
is the law of formation. 
Particular solution, 
«=^„(D)H+e*4.,(D)H+£“,/-2(D)H+..+s-4.„(D)H+.., 
where 
is the law of the formation of the operative functions. 
2 N 2 
