252 
MR. BOOLE OX A GENERAL METHOD IN ANALYSIS. 
Comparing this with the general form of Ex. 1, we have 
a l — i, a 2 = — i — 1, n= 2, q—h, whence 
D — 1 
“= S ~^ P 2D-W-1 V ’ 
d~v 
v being determined by the equation li 2 v—i 
d 
(D — 2/-J-1), and writing x-^ for D, we have 
The value of u may be otherwise expressed thus. Applied to any subject, we have 
D — \ =z e Dz~ 6 —x:( — ? 
\ dx! x ax x 
d 1 
d_ 
dx 
2i+l)i 
(36.) 
D — 3 = eWD&-M=x4- 
dx x 3 ’ 
.d 
D — 2 / — j— ]. —e( 2i ~ i yi)s~ ( 2i - l )t==x 2i -j^x (2i-1) 
Substituting these expressions in the general value of u, viz. 
U = i~ ie ( 
we find 
— 1)(D — 3)..(D — 2/T l)v, 
'■=¥ ■ ■ ■ C‘sL~ <5 ‘~ '"•>> 
__ _L ( x zA \{ x zA\ ( x Z— \ V . 
x l+ 1 V dx'\ dx ) ' ‘ dx! x u ~ 1 
-JL f^sAY 
~x i+, V dx) 
p2i — 1 
Hence the complete integral of the equation ^ — - 1 -- - u -f- h 2 u — 0 is 
1 ( ,, d V ccos(hx) + c, sin (Ax) 
“ =^' V*W ’ 
d i/(z “I - 1 } 
and that of the equation — —^■u—h 2 u = 0 is 
u 
1 / „d\ i Ci 1 >x -]-c l s Ux 
~)d Tl \ x Tx) x 2 i i| ( 3/ *J 
which forms are perhaps new. 
Equations of the above class have been discussed by Mr. Leslie Ellis, in two 
very ingenious papers published in the Cambridge Mathematical Journal*, and it is 
just to observe that the first conceptions of the theory developed in Prop. 3 of this 
section, were in some degree aided by the study of his researches. 
The tw r o following examples are intended to elucidate the theory of disappearing 
factors. 
* Vol. ii. pp. 169, 193. 
