466 
THE REV. B. BRONWIN ON THE SOLUTION OF 
complex. And we may observe that if the functions in (1.), (3.), &c. have 
suitable factors, the reduced equations (2.), (4.), &c. may be still further reduced by 
the same means, the proposed equations in this case being of an order above the 
second. 
From (5.) and (6.), makingy’(z!7)==^(OT) = l, we have 
ziy(z!r-\-k)u-\-p(ziy-\-nk)7^^'u='K (13.) 
and 
{7S-Y{n-{-\)k)v-\-pT'^v=v\^^ X; 
whence 
Similarly, from (7.) and (8.) we find 
{ts ■^nh){rs-\-nk-\-k)u-\-p7ST^^u—l!i. ( 14 .) 
“^^^|x. 
The two last examples, reduced to the ordinary form, are 
-{■2'k-\-k + jor^*) Dm + (^X' + + X ( A: -{-pr^’') + (w -1- 2) kpT^'‘) w = X, 
and 
<p^D^M+<p(^'+2Xd-(2M-!- l)/i:-l-jE)r^*)DM+((pX'+X^-|-(2M-l- \)k\-\-{2k-\-X)pT^''-\-n{n-{-\)J^)u 
respectively. 
I shall only give two other examples, derived from the same source with the two 
preceding. 
'Zo{7S-\-k)u -\-p {7S-\-a){zs-\- nk) = X, 
( OT + nk) {TS-\-nk-^k)u (^tr + a) = X, 
where 
[n + \)k-\-{a-\-2k)pT'^^ k-k-{a + 2k)pT^^ 
1 +pT^'‘ ’ 1 +pT^^ 
II. SECOND GENERAL THEOREM IN THE CALCULUS OF OPERATIONS. 
Make D=Di+D 2 , Dj operating upon u only and D 2 upon x only. Then by T.vylor’s 
theorem 
/(D) =/(D.+D,) =/(D,) + D,/'(D,) +iD|/"(D.) + . . . . 
u 
_pl(^+l)^ 
{zs -\-k-\-pr^^} 'P 
{n 
k 
