280 MR. HARGREAVE ON THE RESOLUTION OF LINEAR EQUATIONS IN 
In general, it will of course be understood that these researches, as to convergency 
and divergency, relate only to the complementary series, or in other words, to equa- 
tions deprived of their second member; nevertheless they will to a great extent, if 
not throughout, apply to cases in which the second member is in the form of integer 
powers of x or of log x. 
1st. When in the set of functions 
/.(D), /(D), /.(D),.../_,(D), /(D). 
the function yoCD) is of a higher dimension with regard to D than any other of the 
set, or is what we shall here call the dominant function, the solution of the equation 
can always be found in a convergent series of ascending powers of x\ and if in such 
a case we solve the equation in a series of descending poweis of x, which we can do 
if we please, that series is certainly divergent. 
This is immediately apparent from the consideration of the law of the coefficients 
which, as ni increases without limit, approaches to the form 
A -_^A - 
•? 
assuming that all the functions are only one degree lower than A^, which is the least 
favourable case for convergency. Therefore, if the largest of the terms of the right- 
hand side of this equation be 4_,A„j_^, we have 
m 
and we can therefore arrive at a point in the series at which the ratio of the coeffi- 
cient of .r”* to that x'^~^ diminishes without limit. 
It will also be observed that the series introduced by two or more equal roots are 
of the same character as the original series from which they are derived ; for, A^ being 
dA dA 
of the form <p((3i-f m), we have ’■> when A^_p is of a higher order with re- 
ference to m than A^, is also of a higher order than and so for the other dif- 
ferential coefficients. 
We have now merely to inquire what must be the form of the original equation 
that /o(D) may be the dominant function. Referring to (2.), we see that it is neces- 
sary that the factor of the highest differential coefficient of u should contain one term 
only. If this factor be 1 or x, no restriction need be imposed on the succeeding 
factors. If it be <r-, the factor of the next lower differential coefficient must not 
contain an absolute term ; and generally, if x^ be the factor of 
d’^u 
dx^^ 
the factor of 
d'^~'^u 
must begin with a term not lower than x^~'-, that of with a term not lower than 
and so on. 
