FINITE DIFFERENCES AND LINEAR DIFFERENTIAL EQUATIONS. 
285 
coefficients of each series is distinct, and substantially the same in form for all ; and 
that it is not necessary to have recourse to the method of parameters in the cases of 
equal roots, or in the case of there being a term G on the right-hand side of the 
equation. In the case of imaginary roots, the laws of the series are in the first instance 
distinct from each other, and afterwards combined in couples. 
19. The investigations contained in the latter part of this paper reduce the problem 
of the integration in finite terms of the general linear differential equation with 
rational coefficients, to the finding of an algebraical expression representing the deve- 
lopment 
+ -}- A^X'"-1- ..., 
where Ao= I, and the law of relation is 
/omA^-f/mA^_i-f-/2mA^_2-|- .. +/.^A„_,=0 ; 
the functions foff 2 '-fr being known specific functions. The series to be summed 
closely resembles a recurring series ; it differs from it in this particular, that the law 
of relation, instead of being constant, has a uniform and simple variation as it pro- 
gresses along the series. If the rational coefficients should be themselves infinite 
series, the process still applies, the only difference being that each term would be 
formed from all the preceding terms, instead of being formed from the r immediately 
preceding terms, or from all the preceding terms when the number of them is less 
than r. 
In those cases in which the equation is soluble in finite terms by known methods, 
we are enabled to assign the algebraical expression for the series ; a result which may 
be used for the discovery of generating functions. 
Thus, taking the general equation of the first order, 
du 
{a, -f- -f + . . + ^ + («o + -f Cox'" + . . + hx’"' * ) M = 0, 
we see that 
_ /*J go + ioJ^+ ■ +A:ot”-1 
where Ao=l, and the general law of the series is 
aimA,„-l-(&i(m— l)-l-ao)A„_i + (c,(m— 2)-1 -^o)A,„_2+...(A(w — w) + ^o)A™_„=0. 
If we take the general equation of the second order, 
d^u dvL 
^ + {a^+b.x-^- = 
the law of the series will be 
a^m(rn—\)A^-^{bJjn— l)(m— 2)-l-ai(m— l))A^_i-l-(c2(//i— 2)(m— 3) + &,(m — 2)-j-Uo)A,„_2+ . 
-\-{k{m—n){m—n—\)-\-k^{m—n)-\-h^)A^.^z=0 ; 
and generally, in equations of the nth order, the law of relation involves the wth 
power of m, the number of the term sought for. Thus the determination of soluble 
