mains to consider 
LINEAR DIFFERENTIAL EQUATIONS. 
469 
<P(D)' 
aD 
1 ^ 
by substituting for its value. Bute cannot be interpreted except when 
;^(D)=§'D, We have therefore, finally, 
m 
r'-‘=yD, ?(D) = -iD, X(D) = -5p,(>«<I>(D)+<I>’(D)) 
This result narrows very much the limits of the practicable cases. 
We may however obtain a small increase of generality by making x;(D)=^B+r. 
There is nothing to prevent this, since 
^-f(?D+r) 
■JEI -Pl-Q 
£ *2 * , 
which makes no change in the interpretation of the results, the only difference which 
_jfir 
it occasions being the introduction of the constant quantity £ * into the value of u 
where we should otherwise have unity in its place. Thus we shall have 
r’-‘=?D+r, f(D) = -i(D+Q, MD) = -/((D+^)(m+'||5|); 
which values give 
By the conversion of symbols we derive from (9.), 
Tz' (zr' + « ) M +J0 (w' + nk) X, 
«=P'{f}(^'+«+;,r'-‘)-p|f}"x, 
which by putting for its value becomes 
z«r'(z3-'+£f)M-f-p(Tir'+wA:)(g'D+r)M=X (16.) 
From (11.) we derive in like manner 
(tz' a) (zz' -j-nk)u-\~p'!!r' (^D =X. (16^) 
M=p|”*^|"‘(^+pyD+pr+a)-P'{[” 
But (10.) and (12.) treated thus would lead to equations of the third order; and 
as we cannot notice those of all orders, we shall pass these by. 
It would render the values of w too long and complex to substitute for zz' its value 
and reduce them further. But this is unnecessary, since the method of doing it has 
been made sufficiently plain, and indeed is well understood. If we would reduce 
these equations to the ordinary form, it can easily be effected by the formula 
/(D) =/(D,+D,) =/(D.) +D/-'(D.) +iDV"(D) +&C., 
3 p 
MDCCCLI. 
