FINITE DIFFERENCES AND LINEAR DIFFERENTIAL EQUATIONS. 
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where Ao=l, and m+l)A„,_,=0 is the law of formation ; the series 
then are 
r 
/3i 
^1-1 
‘■‘I'** “/.(ft - - 1) 2) 
+ 02 { similar function of jSa . • • • 
7.(/3i - 1) /-(A - 2) 7(/3i -3)^ + • •/ 
}= 
which is, {since Jl{^i — n)=n{n-\-l)c 2 — 2 n(^i—nb^), 
r — ^ — — 
2c^-2/3i-l>i ^ 2 c 2 - 2 / 3 i - 6 i 2.3c^-4^,-2b, 
/3, /3,-l /3i-2 g_3 'I 
~2c2-2/3,-6i 2.3^2- 4/3i- 2^1 3.4c2-6/3i-3Z»7 ' 
+ C 2 { similar function of |32 }. 
When (3^ or (Sg is a positive integer, one of these series is terminable ; and if both 
are positive integers, the series derived from the smaller root is terminable, and the 
other gives no result, the coefficients becoming infinite. The first of the series will 
then be found to give the same result as that produced from the divergent series 
(which is evidently terminable in form in the case indicated), except that it begins at 
the other end. In this case the other complementary solution can be found in finite 
terms by reducing the order of the equation. 
The particular solution is 
D 
m=Hi— j Hi+-y 
D 
D-1 
/;(D-i)/,(D-2) 
H-..., 
H. being (/.(D))-^(5-«G). 
Let us now return to example 2, the solution of which, as above found, is in some 
cases divergent. 
The transformed equation now becomes 
(^(D + 2)2+r)M+jo(D + 2)(r®w) + ((D + 2)2-r2)(r^t4) = r^<’G. 
Roots of frt~0 are — 2 +/^/ (^, which call and ^ 2 - 
First complementary series, 
^Ao+^+^4----f-^+--^5 
where 
Ao=l, and (y(|3,-m+2)"+r)A^+jo(^, — m+2)A„_, + ((/3,— m+2)"— r")A^_ 2 = 0 , 
is the law of formation ; and the second complementary series is the same, using the 
other root with a different constant. 
In like manner the particular solution is easily represented. 
16. We are now in a position to discuss the character of the various series obtained 
by this process with reference to their convergency or divergency, a subject of the 
highest importance to the value of the process. The investigations which follow, 
will, it is apprehended, be found to afford a complete test of the nature of the series. 
