MR. BOOLE ON A GENERAL METHOD IN ANALYSIS. 
253 
Ex. 4. Given ?‘+ l)^)^+((a+ \)qx— bi)u=X, the 
second member representing any function of x, and i being an integer. 
The symbolical form of the above equation is 
n 
+?(IOTV«= U > (38 ' ) 
wherein U= {(D + Z»)(D — /)} _1 X with the relation x—i*. Assume as the transformed 
equation, 
D + c 
v + q 
D + b 
e v=\, 
then P 1 ||§j = P l] ^=D(D-l)..(D-/+l), wherefore 
tt=(D)(D- l)..(D-i+l)*;, ' 
U=(D) (D — 1)..(D — /+1)V. 
(39.) 
(40.) 
Now (39.) gives 
(D+6) r+^(D+fl+l)^=(D4-^)V, 
^+bv+qx(x^+a+\)v=(D+b)Y, 
v= 
(41.) 
' x\\ + qxf- b+l 
Now from (40.) we have 
V={D(D-1)..(D-/+1)}- 1 U, 
/. (D+&)Y={D(D-1)..(D-/+1)}- 1 (D+6)U. 
But (D+&)U=(D — i) _1 X, whence 
(D + 6)V= { D(D - 1 ). .(D - i ) } - ' !X. 
In performing the inverse operation {D(D— 1)..(D — /+I)}- 1 we must, by the 
second canon, retain one arbitrary constant. We choose the one derived from the 
r / (l \ 2 1 -* — 1 
factor D. Observing then that {D(D— 1)..(D — j , we have 
(0+^=0)-^+ c. 
Hence substituting in (41.), 
Jdx^-^l +qx) a ~ b (^jy?.dx i+l -~+ Cj^l + C 
x b (l + <p')“~ 4+1 
U X \dx) x b [l 4- qx) a ~ b+l 
If X=0, the above gives 
„ _ r i MV C+C , fdxx b -'{I + qx) a - b 
\dx) x h {\ + qx) a ~ bJrl 
