1916-17.] Operators applied to Solution of Equations. 33 
take D a full time or times, and the left-hand terminal in (3) cannot be 
q or rn . This terminal has, in fact, become indeterminate. The right- 
hand terminal, on the other hand, remains fast, for the only thing we know 
about D B m is that it diminishes the index of any power of B by — , and 
1 1 
thus converts vi+1 into ( r+uw+ i . The last multiplier, therefore, must have 
B m B . ** 
( T* ~ | 1 ^ 11s ~ | 1 
been this new index diminished by 1, or — — — 1, retaining this 
feature of common differentiation if and as we may. We write, therefore, 
1 
n 
provisionally for jya . > 
F 
(r + \)n+l 
1 
m 
(r-Fl)n-Fl 
B 
and for 
(4) 
21 C 
n \m / . ] ) m — 
U c -U B 
r n—rm+1 
to 
rw+l 
B m 
Tj 1 / 1 + 1 ) (r+ l)(n -m)+ l 
1 [ m j C m 
l in 
where c is some possible factor independent of r. 
Putting r = 0, 1, 2, etc., in, we have from (1) 
F J (y+l)(w-w)+ 1 1 
i 
F 
n + 1 
m 
1 
F 
n -7H+ 1 
m 
l_ l 
m ’ c 
F 
2 n 4- 1 
m 
F 
2 n + 1 
- 9 
in 
1 2 n — m + 1 1 
m m c 
F 
(3n + 1 
\ m 
J3n + 1 
m 
1 3 ii — m+ 1 3 n - 2 m + 1 
etc., etc. 
m m m 
From the first of these we see that c = m, and the others are then all 
( l^'Tb ]_ 
satisfied by assigning to F 
'rn + 1 
(5) 
1 
111 
m 
rn + 1 
_ 9 
— 1 j the form 
rn + 1 
m 
-3 
m 
. ad infinitum . 
I leave the reader to verify that, had we solved for x m , we should 
have got 
2 n 
X = — + 
C AO . 2 n A 2 C 5 
B 
+ 
, '-+i m 
B m 
+ 
3n ( 3n 
B 
— +i m \ m 
— - 1 
A 3 C 
Oo 
m 
3! AW 
B m 
j etc. 
• ( 6 ) 
and that this is the work of the same operator Dp.D ( d m . D B m on yu as 
B’ 
VOL. XXXVII. 
