LINEAR DIFFERENTIAL EQUATIONS. 
475 
But if %(D)=^D, we shall have 
■^(0) = — (^am-\-ar-\-cq-\-\a^J)—{cm-\-cr-\-c). 
With either of these values, and by suitably assuming m and n, we may find con- 
venient practicable forms for both the equations (20.). And perhaps we may succeed 
with other assumptions. If we admit complex forms, we may have them in abun- 
dance. 
IV. SOLUTION OR REDUCTION OF ANOTHER SERIES OF EQUATIONS. 
The equations here treated of are generalizations and extensions of one solved by 
Mr. Boole in the Cambridge Mathematical Journal, No. 1, New Series. In that 
equation the coefficients are integer functions of x, here they may be any functions 
of that quantity consistent with the conditions of integrability. Also the symbol D 
is replaced by zs, and arbitrary functions of this last are introduced. 
In order to abridge I shall write for w-f-a, and /(ot) will denote any rational 
function of cr. This being premised, let 
/(w)/(uy — /r)tff(nr-]-/f)M-l-/yOT„(cT„-j-(2w-f-l)A-)T"*M = X (21.) 
Assume 
the common difference of the factors being here 2k. Then by («.) we shall have 
(ni„-l- (2w+ 1 )k)T^’^u= (nj^+Zr) .. . . (ro„-l- {2n -\- 1 )Z:)T'*y. 
Substituting these values and reducing the result, we find 
/( w )/( OT — Zf ) ro ( ro -I- A- ) y + p nr „ ( nr „ -f Z: ) r = P I ^ ^ ^ j X 1 , 
where the a is suppressed in the factorial of the second member for brevity. 
we have 
,, I (-STa-l- A:)'srg 
‘ Z^/(’S7) (ct -|- k)f{z7 — k) *’ 
or 
Make 
V 
/ tSg + k \ CTa + A: 
\/w} (^ + ^) / \/(®') 
'^r'^v=X„ by (a.). 
ji 
If we put 
and the last becomes 
v-\-pfv=Xi. 
The mode of treating this has been already explained. By it the proposed is made 
to depend upon two others, each of which is of an order only half as high. In certain 
cases therefore this reduction amounts to a solution. 
