278 MR. HARGREAVE ON THE RESOLUTION OF LINEAR EQUATIONS IN 
The only complementary function to be obtained is therefore 
*< = Ci| 1 — + 1 «o (^ + «ok“ - ^ «o (^ + «o) ( 1 -2 ^ 2 + 2 + flo) 
+2:^«o(^+«'o)(l*2c2+26iH-ao)(2.3c2+36i+ao)^y^-...}. 
a series which is divergent, but which, as will also be seen afterwards, is finite when 
the constant coefficients are connected by the formula {p— \ )pc 2 -\-phy-\-aQ=. 0 , p being 
any positive integer. 
Further, it may happen that the series obtained by the process, even when they 
afford a complete solution in respect of the number of constants, are divergent for all 
or for some values of x. This may evidently be the case in the solution of the second 
of the examples above given; for the law of the coefficients, as we advance in the 
series, approximates to A^= — qA^_^. 
In these cases, other solutions in series may be obtained by resolving the equation 
in finite differences in a series of terms of the form instead of In order 
to effect this, all that is necessary is to write 0-\-r for 6, (r being the order of the 
equation in finite differences), and to divide by the factor of the last term instead of 
the factor of the first term ; or in other words, we must pass s'"® outside the functions 
in (2.), and multiply by ; so that this equation now assumes the form 
/(D)M+/_.(D)(£-*!,)+/,_,(D)(r*'M)+...=£<-»G; 
and the equation in finite differences is 
fr{0)Ue-\-J'r-l{^)'^e+l -{-/ r- 2 (,^)U 0 + 2 -\- - • .= Gg+,._„. 
We have now to inquire for the roots of frt=0 ; the incipient term is 
which call Hj ; and the particular solution will be found to be 
where 
4/o(D) = 1 , and/,(D - m)'4^^(D) +/r-i (D — 1 (D) + . . . = 0 
is the law of the formation of the operative functions. 
The substitution of the roots of successively for D gives the law of formation of 
the complementary series. 
Taking the last example, the transformed equation now becomes 
(c 2D(D— l)+&iD+ao)M+(D+ 1)(£“®«)=£“^®G. 
Let the roots of or l)-\-hit-\-a^=0, be (3, and 
The first complementary series is 
(Ao+A,a^-‘ + ..+ A^x-’" + . .) , 
