FINITE DIFFERENCES AND LINEAR DIFFERENTIAL EQUATIONS. 
273 
/i(D-l) 
/.(D) 
M„+ 
/i(D-m + l) 
/o(D-m + l) 
M 
m— 
■ /2(D-”^ + 2) 
'^/o(D-m+2) 
/^(D— OT + r) 
/o(D-m + r) 
M = 
0 ; 
or, by passing s®, &c. outside the operations by the equation 
^(D) =£“V(D +m) H, 
t^=MoH+/MiH+™2H+..+£™^MJ^+..., . . . 
where the general law of relation is 
A/T I /i(D + l) p,/r / 3 (D + 2) , /,.(D + r) _ 
/o(D + l) i + /o(D + 2) •• +/o(D + r) 
. ( 3 .) 
Now the expression H, which is the subject of all the operations, contains n arbitrary 
constants, since /o(I^) ^-^e wth order ; that is, provided is not zero. 
Let (3^ (32... (3„ be the roots of ffft=0 ; then the complete value of H is 
(/,(D))-'(a»^G)H-c,a^'^+c/^«+ ..+cy»^ ; 
« 
of which the first term will give us the particular solution of the original equation, 
and each of the other terms a complementary solution. Bearing in mind the equa- 
tion (p(D)(a^®) = <p;>.£^®, we see at once that the first complementary solution is 
c/'^(Ao+A/-hA/^-f..+A„, 
or 
CiJ^^/Aod-Aia^-j-AgJ^^H- . . + A„j.t”-[- ..), 
where Ao=l, and the law of formation of the coefficients is 
/o((3i+yw)A»+/i(^i+^)A^_i-}-/2(/3i-l-m)A„_2+ ..-f/(^i+m)A„_,=0 ; 
and the remaining complementary solutions merely require the substitution of | 82 .../ 3 ^ 
successively for (3^, with new constants. 
The reader will not fail to perceive the peculiarity of the series when a„ is unity, 
(which value it can always have when it is not zero) ; for in that case the roots of 
y‘o^=0 are the natural numbers from 0 to n—1 inclusive, so that the series begin re- 
spectively with the terms 1 , , ... . 
If a„ be zero, that is, if the factor of the highest differential coefficient of u do not 
contain an absolute term, then in order that the transformed equation may take the 
form (2.), it will be necessary to pass £® outside the operative functions in each term, 
and divide by £®. The initial function /"of will then be of the form 
6„(L..(^— w+l))-l-a„_,(L..(^-n-f2)), 
and H will be 
(/o(D))-(s'— '"G). 
Similarly, if in addition to a„ being zero, we have also h,, and a„_i respectively equal 
to zero, it will be necessary to pass £^® outside the operative functions, and divide by 
2 N 
MDCCCL. 
