of Edinburgh, Session 1875-76. 
95 
Also 
d /Pn\ Pn + l | ^ 1 
T V \Y) = Y+'' whence v '~* 
Yd 
dp\p n / p 
Eliminating P w between (5) and (6), we obtain 
\dpj 
0 \dpj 
Ip J 
( 6 ). 
(7). 
This equation is thus true for all positive integral values of n, 
and its form at once shows that it is true for negative integral 
values also. It is very singular that such a series of equations of 
all orders should have a common solution. But it depends upon 
the fact, which I do not recollect having seen in print, that 
\dx ax/ \cLxJ \dxj 
This can be verified at once by applying it to x m \ as can also the 
companion formula 
( d Y fd\* n 
[x — x) = x>h — ) x n . 
\ ax J \dx/ 
Suppose we had, instead of (5) and (6), 
f/Q/i 
d 
dq 
dq 
( q n Qn ) 
_ — Qn + 1 
n— 1 
Q» — ; 
(5 1 ), 
( 6 1 ), 
we should find the same equation (7) for Q 0 as for P 0 . In fact, as 
is easily seen, 
Q, = P». 
Other pairs which alike give the equation 
K, 
are 
and 
dr 
dSn 
ds 
= R 
n- f- 1 
dr 
d / Pn\ _ R » — 1 
\ r n J ~ r n — 1 
(7 1 ) 
’ dr 
d 
— S re _i , ^ s (s n S w ) — s n+1 S n+ i . 
We thus get the two distinct particular integrals of each of the 
corresponding differential equations. 
