MR. BOOLE ON A GENERAL METHOD IN ANALYSIS. 
261 
Let u— z 26, v, then 
n (D' — a") ( D — 13") u — a!) (D 1 — (3') g *u -f- /(D' — a) (D f — (3) = 0, 
wherein D'=^- The comparison of this equation with ( 53 .) shows that if in the 
equations of condition which we have obtained, we change a into —cc", (3 into —(3", 
l into n, and vice versa , we shall obtain a new series of conditions of integrability. 
There are probably a few cases which the above analysis would fail to discover. 
Should the attention of analysts be turned to this subject, it is not unlikely that we 
shall soon be able to tabulate the forms of f 0 ( D),f 1 (D),f 2 (D), which render integrable 
the equation 
/o(D)»+/ 1 (D) £ '«+/ 2 (D) £ « m= U, 
aD object which I have endeavoured to accomplish for the case in which the first 
member involves only two terms. 
D. Theory of Series and of Generating Functions. 
Let UpX p -\-u p+1 x p+1 ...fiufi be the proposed series, and let the law of derivation of 
the coefficients be 
u m +(pi{m)u m _ 1 ...+(p n (m)u m ^ n —0, ( 55 .) 
a law which we shall suppose to obtain for every set of n-f- 1 consecutive coefficients of 
the series. This condition excludes from ( 55 .) all values of m from p to p-\-n— 1, and 
from t-\-\ to tfin, i. e. the n first values of m in the series, and the n first values of m 
following those in the series, because for such values of m ( 55 .) ceases to be a relation 
connecting n-\- 1 consecutive coefficients of the series proposed. Now by the funda- 
mental theorem, if u = ^u m x m = 'tujF^, then 
but by ( 55 ), the expression w m -f<p 1 (m)w m _ 1 +&c. vanishes except for the values of m 
above particularized, hence to those values alone of m is the summation in the second 
member to be extended. The result may be expressed in the following theorem. 
If u — ^u m x m = ^u m z m6 , and if every n-\- 1 consecutive coefficients of the series are con- 
nected by the relation 
u m -\-<p 1 (m)u m _ l ...-t-<p n (rn)u m _ n = 0, 
then 
z«+9i(D)^- + < P„(D)2' ? ^=2{(w OT +^ 1 (m)w m _ 1 ..-{-^( m ) M «-«) £m< '}^ • • (XVII.) 
the summation 2 in the second, member of the equation extending to the first n values of 
m in the original series, and to the first n values of m following those which are found 
in the series, every value of u m being rejected which is not contained in the given series. 
The following are particular deductions from the above theorem. 
Let u=u p x p -\-u p + r x p+rJ ru p+2r x p+2r —-\-u t x t , and let the law of derivation of the coeffi- 
cients be u m =<p(rn,)u m _ r , then 
u m —<p(m)u m _ r — 0 , ( 56 .) 
u — <p ( Ti)i r6 u = 2 { u m —<p(m)u m _ ,.) i m6 } . 
