FINITE DIFFERENCES AND LINEAR DIFFERENTIAL EQUATIONS. 
281 
In short, all equations where x does not enter into the factor of 
d”u 
dx”’’ 
and all equa- 
tions of the form 
1/7 « 1 1 7 « I \ 
■{KxP 
where is a positive integer, are soluble in convergent series of ascending powers of x. 
The third example above given is an instance of this form. 
2ndly. When in the set of functions 
/,(D), /..(D), .../.(D), /(D), 
the functiony’,.(D) is the dominant function, the solution of the equation can always 
be found in a convergent series of descending powers of .r; and if in such a case we 
solve the equation in a series of ascending powers of x, that series is certainly diver- 
gent. 
This is apparent, as before, from the consideration of the law of the coefficients, 
— m) A„_, + . . . 4/o(3, — m) A^_,= 0, 
which, as m increases without limit, approaches to 
l-^m— 1 ••• 
in the least favourable case for convergency. 
On proceeding to inquire what must be the form of the original equation, we see 
again that it is necessary that the factor of the highest differential coefficient of u 
should contain one term only. If this be x^, then the other restrictions are, that the 
factor of the next differential coefficient must stop at that of the next at x^~^, 
and so on. 
In short, for all integer values of p the equation 
xP 
d"u 
dx'"' 
^ u 
+ (««-2 + + • . + 
d^~^u 
dx^~^ "T • • • 
is soluble in a convergent series of descending powers of x. The 4th example above 
given is an instance of this form. 
Srdly. When in the set of functions 
/(D), /(D), .../_,(D), /(D), 
the functions ^ 0 ( 0 ) and /,.(D) are of the same dimensions, and are both dominant 
over all the other functions, the solution of the equation can be found in a series of 
ascending powers of x, which for some values of x is convergent, and for other values 
of X is divergent ; and the solution can also be found in a series of descending powers 
of X which is divergent for all values of x for which the other series is convergent, 
and convergent for all values of x for which the other series is divergent. 
For in the ascending series the law of the coefficients approaches the form, (the 
2 o 
MDCCCL. 
