282 MR. HARGREAVE ON THE RESOLUTION OF LINEAR EQUATIONS IN 
coefficient of the highest power of m in f^m being 1, and that in being /„,) 
A — 1 A 
and in the descending series the law approaches to 
The former series is therefore convergent for all values of x numerically less than 
_ 1 
(4) V, and divergent for all values of x numerically greater than this limit ; and the 
latter series is divergent for all values of x numerically greater than this quantity, 
and divergent for all values of x numerically less. 
The equations to which this rule is applicable are of the forms, {p’ being less 
than p,) 
d^u 
d^~^u 
and 
The second example above given is an instance of the first of these forms. 
4thly. When in the set of functions 
/.(D), /,(D), (D), /(D), 
one or more of the intermediate functions is or are of the same order as the extreme 
functions /o(D) and or as the highest of these two when they differ in dimen- 
sions, the series obtained by the above processes will be divergent for some values of 
X, and we have not as yet any method of deriving a convergent series corresponding 
to these values ; and if one or more of the intermediate functions be of a higher 
dimension than the extreme functions, the series obtained by the above processes will 
certainly be divergent. 
These remaining cases therefore sever into two species ; first, where some of the 
intermediate functions are of the same order as the highest of the extreme functions ; 
secondly, where one or more of the intermediate functions are dominant. 
The first of these species includes equations of the two following forms : — 
in which there can be no function higher than /o(D) ; and 
d^ 
{an^bnX-\- . .-\-lnX^) + + j „_i +... = 0, 
in which there can be no function higher than /,(D). 
In these cases it will easily be seen that the law of the coefficients of the ascending 
series, as m increases without limit, approximates to 
”1“ • ’ • ~1~ ^n^m-p b, 
