PROFESSOR G. H. DARWIN ON ELLIPSOIDAL HARMONIC ANALYSIS. 
475 
If we differentiate in its two forms we find 
fZP^ 
dv 
And 
= — 1 ) 
fZP^ i t\ 
dv i — t\ 
tv 
2) / 
! d V 
dV + '^ 
tv 
p^+1 
P = „T-_r^ P'+ 
V 
\ dv 
\dv 
! d \-< 
- tv{y^ - P + P 
?' + Z ! i — t 11 P^ 1 
-^1 ^ V^t\ ' i + t - 1\ 
1 now write 
{l t) — ("^ + 0 ~ ^ fi“ 1) — ^ ~ !)• 
It is clear that 
{i, — t] = {i, t + ip and [i, 0} = [i, 1] = i{i + 1). 
% [ t ' 0 ~~ t ”j” 1 * 
Now since -— = H, t\, by taking; the sum and difference of the two 
■t — z!z + z — i: L’j’./ to 
fZP'' 
forms of -y-, we have 
dv 
dV‘ 
dv 2{v~ - 1) 
7[F + i + {/, tj P 
'!'-r 
vV‘ 
{V- - If 
_ 1 
- <5 
_ Lpz + i _j_ pz-i 
( 10 ). 
i — t\ 
It is easy to verify, 1w means of the relationship P ' = that these 
equations are true when t is negative. They are als<') true when t — 0, although the 
second equation then becomes nugatory. 
Multiply the first of (10) by v and the second by ^ and a})ply them a second 
time. 
{■?, Z + Tj {0 Z} 2i(Z + 1) 
Then since 
^ + 1 
{•i, Z + 1} 
Z - 1 
{i, t) 
Z^ - 1 
z(z + 1) ^ t{t - 1) W 2 _ 1 ^ ^ ’ 
v- + 1 
dl't 
2v — — L 
dv ~ 2 
^ p^ _ 1 
1 ^ Z(z + 1) 
yi _ 1 1 
P^+y 2Z(i + 1) 
’ zAf-“T Z2 - 1 
F+2 .2i{i + 1) 
Z3 - 1 
- 1 ; 
p/ _p 
P' + 
1 z, Z} {Z, Z _p p^ _ 2 
Z - 1 
{t, Z}ti, Z 1} _ 2 
Z(Z - 1) 
(It). 
These equations are always true although for Z = 1 and 0 they become 
nugatory. 
3 P 2 
