284 MR. HARGREAVE ON THE RESOLUTION OF LINEAR EQUATIONS IN 
17 . Throughout the preceding investigations, the series obtained by the processes 
here displayed undergo a modification of form in the event of the expression or 
fj:, as the case may be, having one or more sets of imaginary roots. 
Let there be a couple of such roots of the form a+|8v^ —1. In the ascending 
series these roots give 
Cia?*+^'^-‘(Ao4-A,,r+A2X^+ .. ..) 
( Bo + B + B#'^ + . . + , 
where 
/o(« + »* + ^\/— l)A;;j_i+ ...=0 
and 
/o(«-f )B^+/(a+m— /3^/^)B„_i+ ... =0 
are the respective laws of formation. 
It is apparent, therefore, that if A„ be of the form ?>(a+(3^— 1), B^ is of the form 
1 ). 
Making C 2 =Ci, and remembering that 
cos (j3 log^) 
—1 sin (|3 logo?), 
the sum of the two series gives the double series, 
2ci.2;“^cos (/3 log a?) { A 0 +B 0 + (Ai+Bi)a:+ .. H-(A^4-B,„)a:’”+ ..} 
sin (/31oga?){Ao— Bo+(A,-Bi)^+..+(A^-BJa^’"+..}) ; 
which is necessarily real, since A^+B^ is purely real, and A^ — B^ is purely imaginary. 
Making now Ci = — Cg, the sum of the two series gives another double series, (making 
],) 
2kx“(^ — 1 cos (13 log x) { Ao— Bo+ (Ai - B,)a?+ . . +(A^— B ..} 
+ sin ((3 logx){Ao+Bo+(A,+Bi)a?+.. + (A^-l-B^)<r'"+..}^, 
which is likewise real. 
The descending series may be treated in a similar manner. 
18. Most of the examples to which the preceding processes are applied have been 
taken from the paper in the Philosophical Transactions for 1844, in which Mr. Boole 
developed his new General Method in Analysis, with which the subject matter of the 
present paper is closely connected, though the methods exhibited are distinct ; unless 
indeed it should prove, that the interchange of the symbol of operation and the inde- 
pendent variable, and the general relation exhibited by Mr. Boole’s fundamental 
theorem of development connecting any system of linear differential equations with 
a corresponding system of equations in finite differences, are merely different repre- 
sentations of a part of some more general method or process. 
The principal difference in results, so far as concerns the solution in series of linear 
differential equations, appears to be, that in this paper the law of relation of the 
