1188 
MR. R. F. GWYTHER ON THE DIFFERENTIAL 
Or, 
[m -|- l) — x) + ^2(1-711^ = 0, 
(m + 1) (tt — p) + ^^3 d-m^ — Oj 
and from these we again conclude 
{iR -f" (I— “h ^2 {d— TTi^q + ]} ^ 
(w “j“ l) d — “1~ 113 id — - 0 
From (20) we deduce that 
- {q + 1) n,} cl>, = 0 .(24). 
That this is an identical relation among the coefficients appears from considering 
the operator. 
- 0 + 1 ) «3i 
which 
= [HjOj — S' flF 
= (niajPi — sOgOj] np-i 
= — q Hg] 
= Hj — (s ~ 1) ^3} 
and ultimately 
But 
&c., 
= ay+^a^. 
^ 2 ^ 0 '— 
nence 
{BjOi — ((2 + 1) ag} (j)q — 0. 
There is then no further condition to be satisfied among the coefficients in the 
expansion. The importance of this will be seen later in the development of 
CO variants. 
12. The order of derivation is symbolised by a chart of the coefficients, wfith arrows 
showino- the direction of derivation. Thus :— 
O 
d^O -^ d^f -^ ^ • • • 
- >■ 0 - 1 ^/ - > d~i^ 2 
^ d-2*Pl 
d-2^. 
d-29 0 
