MR. BOOLE ON A GENERAL METHOD IN ANALYSIS. 
263 
1 d 
m da 
m 
1 d 
m da 
Um — \ — 0. 
whence 
■ U — 
-(D) 
II 
i 
d n ${a) e 
da' 1 1 
du 
d 
t d "$ a 
z n6 
df 
da £ 
U ~~ du 11 
1.2..//— T 
du 
du 
d"$(a) 
x n ~\ 
dx 
da~ 
da' 1 1. 
2. .11 — T 
a partial differential equation of the first order, of which the complete integral is 
u 
x da 
d"$(a) x n ~'dx 
da n 1.2 ..n — 1 
=«**(/«- 
d 
*TaC xn ~ Xdx d ’ 1 \ , i, , N 
= ! J I.2.; r- 1 A r P(«-»)-K(«+*), 
denoting an arbitrary function. 
Now a vanishes with x whatever may be the value of a, therefore the arbitrary 
function and the lower limit of the integral are each 0 ; wherefore 
_ fir , J jm\_ d^a-x) ( 60 .) 
11 J o ax \.2..n — 1 da 11 
d 
the symbol s implying that after integration we are to change a into a-\-x. 
Series of the class f(p ) xp -f- f(p -\-r)xP+ r -\-f(p J r 2r')xP+ 2r -\- &c., wherein f(m) is a 
function of invariable form, may be reduced to linear equations with constant co- 
efficients. 
We have u =f(p) xP J r f(p-\-r)xP+ r -{- &c. Here u m =f(m), u m - r —f{m—r), hence 
wherefore 
_ ,/M 
Myyi J. ?/ „.\ Myyi — y, 01 XI < 
f(m—r) m r> 
u - j(\^lr f 6u =/( 
( 61 .) 
Assume v—z rS v=\, then Pr|^=Pr^^^=/(D), whence 
u=f(D)v, 
f(p)zpt=f(D)V. 
p6 
The last equation gives V=s p 6 , therefore v = - — -, and 
p6 
u=f(D) ‘ 
1 _ 6 r 
(XVIII.) 
This remarkable result may be otherwise obtained; thus, 
u = ‘f (m)e ml >=Jk m= "/(D)s^, 
m = n t/ v y 7 W=p t/ x 
j)6 
=Amz;*-=m ^ 
2 M 
.r6* 
MDCCCXLIV. 
