FINITE DIFFERENCES AND LINEAR DIFFERENTIAL EQUATIONS. 
283 
and the series therefore approaehes without limit to a recurring series, in which the 
constants of relation are 
a „ 
- /yi -- • 
•A/ A cl/ • t • • (A/ « 
dn 
and the denominator of the rational fraction, to which the residue of the series ap- 
proaches, is 
“n 
In the descending series, the law of the coefficients approximates to 
I ^n^m—1 I • • ^n^m—p d, 
and the series approaches without limit to a recurring series, in which the constants 
of relation are 
kji 1 1 dfi 1 ^ 
l„ x’ ■' In XP~'^’ In XP ' 
and the denominator of the rational fraction, to which the residue of the series ap- 
proaches, is 
+ • • + ■ 
When /’o(D) is higher than /r(D), the ascending series alone can be used ; when /r(D) 
is higher than /o(D), the descending series alone can be used ; and when ^o(D) and 
/,(D) are of the same dimension, either may be used; and the approximations above 
referred to render it probable that these series, notwithstanding that they may be 
divergent, are the developments of continuous algebraical expressions. 
The second of the species above referred to includes all equations which are ex- 
cluded from the preceding forms ; that is, all forms which transgress both the restric- 
tions to which the equation in the third case is subjected. 
Of these forms, the solutions, whether obtained in ascending or in descending series, 
are always divergent ; and the divergency appears to be of an extreme and unmanage- 
able character. In this case we have an intermediate dominant function ; and the 
convergent solutions might, from considerations of analogy, be presumed to be series 
infinite in both directions, the roots of the dominant function determining the inci- 
pient terms. 
The treatment of these forms requires the solution of the equation in finite dif- 
ferences 
not starting from either of the two extreme terms, as is done above, but from the 
term so as to get a result in the form 
It would probably not be difficult to show that such a solution exists ; but I have 
not found one in a form available for the purpose to which it is desired to be applied. 
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