PROFESSOE G. II. DARWIN ON ELLIPSOIDAL HARMONIC ANALYSIS. 
Therefore 
- 1 
i/5' 
(I-- )- 
-Jp + 1 
It is obvious on inspection that we cannot rely on this development if s is less 
than 4. 
If then s is equal to, or greater tlian 4, this value of Pq when properly developed, 
to the adopted order of approximation can ordy diher from |[3’ l)y a constant fiictor, 
say Cf or shortly O'’; so that 
= O'P*.(31), 
and we have to determine the constant Ct 
We might develop tlje above expression for P'' completely and compare it witli 
liut this is unnecessary since the comparison of a single term suffices. 
I now write 
.,^5 _ G i + 1) 
~ s- - 1 
or shortly This notation is introduced l)ecause tiiis function occurs very frequently 
hereafter. 
We have seen in (II) (slightly modified) tliat 
p, ^ r p. + 2 
G - 1 ~ ^\s{s 4- IJ 
+ ])P'- + 
(o-sj - 1} 
.s(s - 1) 
We may write this 
where 
Then 
p. 
ifmWl 
1 
a. 
4.? {s + 1) 
p. 
(G-ly 
= a,P-'-’ + /3,P'-'-hy,P'-q 
4.s (.s’ - 1) 
+ /3i (a^ P® " + /3s P' + y.5 P'^ “) 
-f Ys (tts _ 2 P" + - 2 P' “ + 7s - 2 P^ ^)“ 
Therefore the coefficient of P' is a^y^ + 2 + {/^sY + . 2755 oi‘ 
1 { pg + 2} {hs + 1} ^ 1 y: _|_ {hg'l {'0-^ ~ 1} 
] 6 
s{s + 1)2 (.S + 2) 
S{S - 1)2 (.9 - 2)_ 
