MR. BOOLE ON A GENERAL METHOD IN ANALYSIS. 
251 
Here Uj=2, a 2 ~— 3, n— 2; also the equation for v is 0, whence 
t>=csin(^+c 1 ) ; and by (34.), 
-2m D “ 1 
u—z 2, F 2 jyI^ v > 
= g-2^(D— 1)(D — 3)y, 
1 / d 2 d \ 
=?v^- 3r s+ 3 ; csin (?- r + c i)> 
= c {(l i— ?')sin(f»' + fi) — —cos^+c,)}- 
The above example might have been treated directly by Prop. 3, and without the 
aid of Prop. 2, but the final determination of u would not have then depended on 
differentiation alone. Thus we should have had 
u J ^ z 2$ u — 0 
W ' (D + 2)(B — 3j U ~ V ’ 
v +D^=Tf u = V - 
Here ?(D) = (D + 2*(D-3)’ ^( D ) = D(D-1)’ wl,ence P 2||]^ = Lp + ( 2)(D-3) =: DT2’ 
wherefore by (30.), 
U : 
D — 1 
: D + 2* 
„ d-i T7 
°~ D + 2 V - 
As no factors disappear in -^(D), no constants are to be added in determining V, 
whence V=0, t^csin^r+Cj), 
M=^q^ 2 ; =( 1 — 3(D+2)- 1 )csin(^+c 1 ), 
=c(l— 3s -2 *(D) -1 g 2/ )sin(§ , .z+Ci)j 
= c(l-J- 2 (*£) ^sin^+Cj), 
= c{sin(^.r+c 1 ) — f dxxsm(qx-\-C])} , 
= c { ( ! ~ ^2) sin (^+ c i)+^ c °s(§ r ‘ r + c i)}- 
Ex. 3. Given the equation — v — u^rh 2 u=0, i being a positive integer. 
This equation, under a slightly different form, has been discussed by Mossotti in 
his memoir on Molecular Action. It has also been treated by Paoli and by Plana. 
The symbolical form of the equation is 
, A 2 
u 
— (D+i)(D — i— 1) 
z^u =0. 
(35.) 
