MR. BOOLE ON A GENERAL METHOD IN ANALYSIS. 
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parameters, we have 
2{/.(D)s-'(AP+BQ-)}=0. (10.) 
to determine the general character of which let us first consider the term/^(D)s^AP. 
Separating the factors s"'A and P, if we expand the operative symbol f n (D), as in 
Leibnitz’s theorem, we have 
/ i (D)£«'AP=/„(D) £ -'AxP+/,(D)^AxDP+/'„(D) S ”'Ax^D2P+&c. . . (10'.) 
The general value of D*P may be thus ascertained : 
(D)*P=(D)*{/ 0 (D)}-iO, 
= {/o(D)}- 1 (D)^0, 
= {/ 0 (D)}-iO, 
= LP+MQ+NR..., 
L, M, N.. being arbitrary constants. In the present instance, as P does not involve 
any arbitrary constants, and as the direct operation (D)* cannot introduce any, it is 
evident that L, M, N are simply numerical coefficients. 
The above expression for (D)*P applying to every term of the second member of 
(10'.), it is obvious that f n {D)z n6 AP will be a linear function of P, Q, R.., whose coeffi- 
cients are of the general type p n (D)e"*A, <p n { D) denoting a rational and integral com- 
bination of (D). In like manner, ^(D^BQ will be a linear function of P, Q, R, the 
coefficients whereof will assume the form ^(D) s^B. Wherefore the equation (10.) 
will become 
A'P+B'Q+C'R..=0, (11.) 
every coefficient A', B', &c. being of the type 
2{^(D)a^A} + 2{^(D)^B}+&c. ; 
and it is to be remarked, that the terms in this expression which correspond to a par- 
ticular value of n, are derived from the term which answers to the same value of n in 
the primitive equation (8.). 
In order to satisfy (11.) independently of the particular values of P, Q, R, let us 
assume 
A'=0, B'=0, C'=0 (12.) 
Each of these equations being of the general form above given, we shall have by 
the fundamental theorem, 
A =2(a w g M# ), B = 2.(b m s mi ), C=2(c m e m<> ), &c. 
the successive values of a m , b m being connected by a system of relations of the general 
form, 
2 ( <p n (m) a m _ „) + 2 (%/,„ (m) b m - n ) . . . = 0. 
To find the lowest value or values of m in a m , b m , &c., we must assume a m -i, b m - 1 , 
&c. to vanish, whence the last equation gives 
<p 0 (m)a m +$Q(m)b m ...=O. 
