50 
MR. HARGREAVE ON THE SOLUTION OF 
In a manner in all respects analogous, it may be shown that the integral of 
( a^x + + . . + + («o + ^o) = 0, 
is u^=c^J"_^[ay-{-..+a^v-{-aQ)-^vi{v-af'(v—^)^...{\-]rvy-^dv 
+&C. &c. 
If the expressions (1 should have two 
B 
or more equal roots, we shall obtain factors of the form as in the case of linear 
differential equations of analogous forms’. 
The process of changing the symbols maybe used to obtain solutions of differential 
equations from known solutions of equations in finite differences. 
The solution of the general equation of the first order 
(i+ak-p,w,=q. 
IS 
^...x x—2^ X/ 
A similarity existing between this form and the solution of linear differential equa- 
tions of the first order will be seen, by writing the above equations in the following 
form, — 
— (pX.U = 'K, 
j2log(p^'2^g-21og?>(«+l)X) . 
and the conversion of symbols would give 
log?)(D+l) 
a symbolical solution apparently incapable of rational interpretation, at least in finite 
terms. 
If however we suppress X, and by a conversion similar to the one before proposed, 
<pD be changed into <p{—z) ; and x be changed into — D', D' denoting differentiation 
with regard to 2 : and a factor s"*’' be introduced, we shall find that the substitution 
7-^^rY is 'K' ’ general result is 
where 2 denotes summation with reference to 2 $, and the first summation must be 
taken between proper limits. 
The form of this is 
M=2(...(p(w)...<p(— 2 : — 2). (p( — Z — 1 
between proper limits ; which, changing* the sign of may be more conveniently 
'vritten 
