MR. BOOLE ON A GENERAL METHOD IN ANALYSIS. 
267 
In vvhat follows we shall Suppose that the first index p is 0, and that the n arbitrary 
coefficients of the development of u corresponding - to the n arbitrary constants in the 
solution of the equation of differences, are u 0 , u l .:u n _ l . 
Let fim) — t m , then 
U m + Pi ( m ) U m - 1 • • • ■ + P„( m) U m -n “ ** = 0. 
Let also t be the generating - function of t m , as is u of u m , so that 
u=2 m= :(u ^).t=2 m= :<:t 
m=0 \ m ' ni = 0 \ m / 
By the fundamental theorem of development, 
u + ^(jyyu. . -f (p n (py 6 a—v = 2 { (u m + ^Wvi... 4- P n (.m)u m _ n — 
Now considering the expression in the second member under the sign 2, let m — 0, 
and it becomes u 0 —t 0 , for which as u 0 is arbitrary, we may write c 0 , an arbitrary con- 
stant. Secondly, let m = 1 , we have (u l -\-(p 0 (])u 0 —t 1 )e <> =cy, since u Y is arbitrary. In 
like manner may we proceed till we arrive at the assumption m — n — 1 , which gives 
the term c n _p {n ~ 1)6 . For all values of m greater than n— 1, the expression under 2 
vanishes by (66.), wherefore 
u+<p l (D)e <, u...+cp n (P)& n<, u — t=c Q +cy..-\-c n _ l z {n - 1)6 -, 
or replacing t by its value, and transposing to the second member, 
M + p 1 (D)g'M..+p f ,(D)s B ' u^XnZA^Y ' 6 - + c 0 + c/ ..-\-c n _y VJ . . . (XIX.) 
yyi/ Yb — "j_ kyl 2 Kb 2 
Ex. 1. Given u m +a x — — — u m _ x -\-a 2 u m _ 2 =f{m) to find the generating func- 
tion of u m . 
Here by the theorem 
D — n— 1 D — 2n— 2 . . 
u -{- a x — D — i 6 u + « 2 g — | 2 'm = t + c 0 + cy, (6/ .) 
wherein t=K^Z™f{m)z mi . Hence 
D u + « 1 £ ( ’(D — n) u + a 2 s 26 (D — 2n)u = ^ + C A> 
dt 
du a , + 2 a g x dx 1 
dx 1 + ape + ape 1 1 + ape + « 2 ar 
f f y 
“=(I +w+v i T\J (1 2,^+4^ +c l (68 - } 
The value of t will of course be found by the preceding chapter. Suppose as a 
particular illustration that fi m ) = Y2~m ^ ien *’ w ^ ience 
u = ( I + a.x+a.^r + „ , 1 7+ C L»)- + C } ' 
Let f(m)= 0, and further, let C 1 = 0, then 
u — c{ 1 -| -a 1 x-\-a 2 x i ) n ; 
