742 
PROFESSOR BOOLE ON THE DIFFERENTIAL EQUATIONS WHICH 
Combining this with (5), we find 
1 /« = l D+ »\ r /D_»\ 
\ n n) \n n ) [A, 
iW+l) 1” 
e d + A 2 e 29 . . + A n e 
\ + e n 
(AAd+-'] 
y n nj 
l r i 
(D m\ 
l n nj 
'.-.J 
f A,e 9 + A 2 e 29 . . + A n e nd \ 
r(D) i 
[ 1 +e ne j 
u — Q>+ ■ 
=C 0 +. 
Now, resolving the rational fraction, we have 
T ^_ 1 A/ + A 2 e 20 ..+A„e» 9 TW f N^ 9 N 2 e 9 N„e 9 ) 
D 1 +e n9 ~ D ll—u 1 e 9 'l—u 2 e t> ”~t~l-a n e 0 j 
=B 1 log(l — a x e 9 )+B 2 log(l — a/).. +B n log(l —a n e e ), 
■N; 
Hence 
a 15 a 2 , . . u n being the nth roots of —1, and B* 
■ (Bj log (1 — a^ 9 ) . . +B ra log (1 — u n e 6 )}. . . (6) 
r/^D+™W---^ 
■«_n . V 71 nj \n nj 
° 0+ — F(D) 
In this expression B 1? .. B„, being generated from the arbitrary constants C 0 , C 1? .. C„_,, 
may themselves be regarded as arbitrary constants. And this being done, C 0 will become 
a dependent constant, the form of which it will be necessary to determine. 
First, however, let ns endeavour to interpret by a definite integral the symbolic 
function of D. 
We know that a and b being positive quantities, 
!»). 
F(<? + b) 
dt t a ~ 
If we employ the second of these forms, we shall have 
r/'5=*D+”W---i 
\ » n) \n n J 
F(D) 
by a known symbolical form of Taylor’s theorem. Hence if 
t n = T 
1 +* ’ ‘ 
^Co+B, dtt » log (1 — « ; Te 9 ) • • +B„i dt log(l— c5„Te 0 ). . . (7) 
do Jo 
we have 
