MR. BOOLE ON A GENERAL METHOD IN ANALYSIS. 
279 
F. § 3. On the Solution of Equations of Differences in Finite Terms. 
_d_ _d_ 
We have in the preceding section supposed ir=x—xs dx , g=xz dx , and have thus 
satisfied the relations (93.), (94.). We shall, in the following section, with greater 
generality assume (A. Prop. 3.), 
d d 
ir=x — n(pix)s~^, §=<p(x)z~^. 
The condition (93.) will still be satisfied, and the expansion of f{ft-\-g) will remain 
unchanged. 
We shall suppose the equation to be placed, as in all ordinary cases it may, under 
the form 
1K-2+& c -=U, . . . (102) 
ffx),f\(x), &c. being rational and integral functions of x, or at least susceptible of 
expansion in ascending powers of x, and <p(x) any function whatever. 
d d_ 
Since cp{x)f^=^ we have <p{x)<p{x— l s )s~ 2 <^=f } and so on ; wherefore, writing u for 
u x , we have 
/o (f u +ff v )s u +/ 2 (x)? 2u + &c. = U . 
Now since < r=x—ng, we have x=7r-j-ng, and expanding the coefficients f 0 (x) ffx), 
&c. by (I.), the equation wall assume the form 
(pffu + 1 pfF^u + (pfffu -f &c. = U (XXIV.) 
This equation is integrable in several cases : 
1st. If by any determination of n the equation should be reducible to a single term. 
Suppose that it should give 
<p 0 (f)u=V, (103.) 
then resolving <pfx) into its factors, we shall have a system of equations each being 
of the form 
fr— a)u=XJ ; 
or (x— a)u x — n<p(x)u x _ x z=\J , 
which is completely integrable. 
This method enables us to integrate all equations of the form (102.), in which 
!)), (104.) 
h being any constant. We should find n= — h, ir=x-{-h<p(x)e dx , and ff‘7r)u=XJ. I 
am not aware that this general class of equations has been considered before. A 
method of integrating equations of the form 
x(x— l)..(.r— n-\- 1) — 1)-(^ — w+2) A” _1 m^,_„ +1 + &c. = 0, 
was communicated to me by Mr. Gregory*. In reality however they constitute a 
_d_ 
particular case of the above, the value of w being x—xz dx . 
* Late Fellow of Trinity College, Cambridge, and author of the well-known “ Examples.” Few in so short 
a life have done so much for science. The high sense which I entertain of his merits as a mathematician, is 
mingled with feelings of gratitude for much valuable assistance rendered to me in my earlier essays. 
2 o 
MDCCCXLIV. 
