LINEAR DIFFERENTIAL EQUATIONS. 
471 
But we have found 
{ 1 +wD,/(D0+ 
Operating with both members of this on 
{<p(D).r+‘^(D)+mX(D)}w, 
we find 
2”-^(°^{^(D).r+Y(D)+/nX(D)}z< = {?)(D)^+^(D)+mX(D)+n?)(D)/'(D)}s”^''^’w, 
dropping the mark under D as no longer needed. 
Make 
then 
/(D) =/; 
A(D) 
f(D) 
/(D)=^.P(D)/'(D)=MD). 
Substituting this value of /"'(D) in the last equation, the result will be equivalent to 
Change u into and transpose the members, and we have 
Therefore, as before, we shall have 
• («^-) 
By the interchange of the symbols x and D, the two general theorems (c.) and {d.) 
may be converted the one into the other, s at the same time being changed into £“*. 
Before we can employ these theorems in the way intended, we must find the rela- 
tions between the arbitrary functions required in order that two others may subsist. 
The first of which is 
=7!'„7r^M-l- {n - m)a^u (e.) 
With the values given of and ^r„, we easily find that 
t^'tTuU = + (n — m) (p'hlu . 
Therefore we must have 
a§=(p'd, or as'''^'^=(px'. 
Passing to the logarithms of both members, 
log a-irj'^dxzzz log ((pX'), 
and by differentiation 
~dx=—~jr‘ 
<p (p\ 
Hence we find successively 
Kk'dx=d{(px'), -^d{}^) — d{<p'h!), and -= — 
which is the required relation or condition. 
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