48 
MR. HARGREAVE ON THE SOLUTION OF 
or 
ZSZ-={(pz)~'^Z'' 
being the same process as that by which, in the symbolical solution, the form of the 
function of D preceding the factor a:"' is obtained. 
VI. Linear Equations in Finite Differences. 
If in (3.) we change the form of the functions by writing sP for D, we have 
sc(p(s°)u-\-'\p(s^)u='K, 
or x(p{E)u^-\-^{E)u^=Q^, (11.) 
where E is equivalent to 1 + ^? or denotes the operation which is performed in chan- 
ging into u^+,. 
It follows that the symbolical solution of (11.) is 
where it i'=v. 
A case obviously interpretable is that in which •^v=niv(p'v; and in that case the 
solution of 
x(p(E)u^+mE(p'{E)u^=Q^ 
If we take the equation 
(a^x + ^„) . ... + {a^x-\-hi) 1 + {cIqX hQ)u ^ = Q^, 
, fb,v^+...+b^v + bQ J /*/&o 1 , Aj , Ag , \ , 
we have ’-/p = , ; — \dv— (— -+ — —+ — ^+..)dv, 
^ J v\ajif-\r + J V«o ^ v—a. * v—p ' / ’ 
where a, |8, &c. are the roots of ajf-E..-\-a^v-\-aQ-==-0. 
and 
w^=(fl^E”+..-l-«iE-fflo)”'E^(E— a)^'(E— ^)^....{ j 7-^(E~4(E— a)“^‘(E— 
In like manner, if 
^ + • • + + ^l) -f ("O'® +h)u^— 
we have 
where 
b^v^ + ..-^b-{v + bQ — I 
(1 + i;)(a„z;"+ .. +«iV + «o) l+v'^v—u~'v — ^'’'‘ 
Thus the solution of 
x(E2 — c2) — 2mEu^= Q^, 
2m 
or 
IS 
u^+2—-zrUco+i—c^u^ 
Q.r 
^ ^ 1 f ^ ml 
(E— c)~^~ {j?-'(Ed-c)“^(E — c)^Qj} ; 
