MR. BOOLE ON A GENERAL METHOD IN ANALYSIS. 
255 
Iii each of these cases (44.) is reducible by (XII.) to a system of equations of the 
first degree. 
If for ?n l we choose the least of the quantities m l m 2 . . m r , and for n x the greatest of 
the quantities n x n 2 . . n r , the final derivation of u from v will be effected by differenti- 
ation. 
The only integrable forms of the equation u-\- <p(D)z r6 u=XJ , which are not comprised 
in the above generalisation, are those which finally depend on a transformation 
affecting the independent variable. They are included, so far as I have been able to 
ascertain, in the two following general cases, viz. 
, (D + to) 2 + ? 2 TT 
— wnT — \z- s u— U 
1 D + J! D + », 
(45.) 
. _i_ (D + w)(D + ^ 1 ) 
u -r a (D + mf + (f) 
£%= U, 
(46.) 
wherein n — m is an even, and n x — m an odd integer, positive or negative. 
If in (46.) we assume 0=—Q l , then multiply by z 26 \ and reduce, the result will be of 
the general form (45.), which alone therefore it will suffice to consider. 
Bv the successive application of Propositions 2 and 3 the equation (45.) may be 
reduced to the form 
(D — 2) 2 
»+ a D(D-I) ? ^=Vi 
and this equation may always be integrated by putting x in the place of s*. and then 
dx 
^====y, vide (23.). A single example will suffice. 
d 2 u 
du 
Ex. 6. Given ( 1 — (2m+ \)x-^— (m 2 — q 2 )u=0, m being an integer. 
The symbolical form is 
(D + m — 2) 2 — q~ 
D(D-l) 
z 26 u — 0. 
( 47 .) 
Let u—z- m6 u l , then by (XIII.), Prop. 2, 
u 
(D-2 'f-q 2 n 
1 (D— m)(D — m — 1) £ t(l 
Assume 
. 9)2 ffi 
(D- 
D(D-l) 
C t .26 
(48.) 
He" P 2 ^=' ) 2 (lJZ^^r ) =D(D-]) . . (D—+1). whence 
( d\ m 
Mj— D(D — 1) . . (D— v. 
d^v di) 
Now (48.) gives (1 ~^ 2 )^ 2 —< 2 ’^--|-g 2 y = 0, which, integrated by the method above ex- 
plained, leads to y = c 1 cos(g f sin .r~ 1 )-l-c 2 sin( 5 ' sin _1 jr), whence finally, 
/ d\™ 
{c 1 cos(< 7 sin _1 .r)-l-c 2 sin(§ , sin _J x)} (49.) 
2 L 
MDCCCXLIV. 
