LINEAR DIFFERENTIAL EQUATIONS. 
467 
This formula will be of frequent use in the subsequent part of this paper. By it 
j*/(D)_gfc/(DU + *D2/'(D,) + ..._gi/(D,)g*D2/'(Di)+ = 1 1 _j_ 'kT)^f (Q 
Operate with both members of this upon {<p(D),j7+A(D)}m, and we have 
dropping the mark under D, as being no further needed, D everywhere now operating 
upon all that follows it. Make 
/<D)=/,W 
then 
and the last equation becomes 
Now let 
/ ‘riP 
which gives 
1 1 <? t ' . 
^)— 7 <^D’ 
and the preceding becomes 
r' V w = (z3-' + ^) . 
Change u into r'~hi, and there results 
r'^7s'r'~hi = {ts + k) u, 
or 
(7v'-\-k)u = ~ 
by transposing the members. 
From the last equation, by the same course of reasoning by which (a.) was esta- 
blished, we find 
r'-^f{w'-\-k)u=f{ro')7'-^U {h) 
If in (a.) we change x into D and D into x, and also s into we convert (a.) into 
(b .) ; and the same conversion of symbols will change {b.) into (a.). 
Application of the preceding Theorem. 
The equations 
f{Tff')rs'u-\-pf,{m'){7s'-\-nky-^=u='K, 
f{vs'){zs' -\-nk)u -\-pf (nr') wV = X, 
f(nT') Zj' (zj' -I- k)u -\-pf, (nT')(zs' -\-nk)7'~^''u=yi, Sec. 
by the assumptions 
u=(m'-\-k) ....(zj'-\-nk)v, 
U=(zi!' -\-k)~^ (ro'+W^)~V, 
w=(ot'-|- 2A:) (c7'-f-(w-l-l)^)i’j &c*. 
