EQUATION AND ITS TRANSFORMATIONS. 
805 
whence 
1 d V +1 
1 d\‘ e (ix 1 / „d\ i+1 & 
x dx) 6 C \x dx) x x~ i+ \ X dx) x Tl x ii+ ^(lclx) « 3i_1 ’ 
The complete integral of (1) may therefore be written also in the form 
a=AlA-Y (°rL±ML 
x i+ A dx 
1 
or in the form 
1 [ z d\^ +1 (c l e ax -\-c i e- ax ' 
y>i + 3 
dx 
(29) 
(30). 
The first of these solutions, viz. (29), is that given by Boole in the ‘ Philosophical 
Transactions’ for 1844,* and in his ‘Treatise on Differential Equations,’ chap, xvii., 
Boole’s process is as follows : he shows that 
u=e- w (D-l)(D-3) . . . (D—-2f+ l)v 
where 
D denotes 
d 
Id ’ 
x=e, v—~ -— 1 — 
and he thence deduces that 
u=e~ i0 .e 9 De~ 9 .e 39 De~ 39 . . . e (2i - 1)e De~ (2i - 1)9 v 
9 (e 29 D ye~ w ~ 1,0 v 
... 1 ( „ clV[cx ax + c„c ax 
— o (2 ,z l)*Li- 1 —i i 4 - 2 - 
= e~ 
d +l \ cl 
Si -1 
But if the factors are written in the reverse order, we have 
u=e- i9 (D-2i+l) . . . (D-3)(D— i)v 
= e- id .e {9J ~ 1)9 De~ (2i - 1)9 . . .e 3(r D e~ 39 .e 9 De~ 9 .v 
=e (i+1)9 (e- n ~ 9 Dye~ 9 v 
=x i+i(l lYfc^+w- 
x dx 
which is (19). Boole does not seem to have anywhere alluded to the connexion 
between his own form (29) and the form (19), or to have remarked that the latter was 
obtainable by his own method. 
Putting b= — 2 ci {q unrestricted) in (27) and (28), we find 
* “ On a General Method in Analysis,” p. 252. 
