OF EQUILIBlilUiM OF A KOTATINO MASS OF LK^UID. 
323 
Consider the inequality 
(^^o) ^ ^ (^o) n , o ,• , 
ly(z^) ^ ^ ^ ^ ^• 
If the inequality is determined for any value of p, it is determined for all values. 
Now Avhen p is very large 
2il 
PdW = 
p\ 
2il 
2‘ il i — s — 1 
2‘ il i — si 
Hence our inequality becomes 
(i-.srPd(.o)> = <pru-o)- 
This inequality is of the same kind for all values of p^). Now P/involves the 
factor {pff — 1)^® and P/'*' ^ (vq) involves (t-y — Putting therefore 1 + e, 
the left-hand side involves and the right It follows that unless 5 is equal 
to i the left-hand side is greater than the right; but 5 is necessarily equal to ^ — 1 
at greatest. 
Therefore 
PF (pq) PcyiyM 
IV C) ^ PF+Vv' 
Hence K’s with smaller 5 are less than those with greater s. 
It remains to discriminate between the two sorts of P-functions which occur in 
ellipsoidal harmonic analysis ; that is to say we must determine 
^ Pf (^o) 
W(y pfo * 
• fC 
Since the /8 of “ Harmonics” is equal to .-;;; in the ])resent notation, when /3 
and K are small we have by the formulse of that paper 
py(.)-f \K%,.vr^{p) + \K%_.vr~{p) +..., 
p,‘M = 
(■'- - i/«n> 
-1)* 
r'(>') + yr'<A.«P"F>') + 
_ 0 
When p is very great and k very small p/ = P/, so it suffices to determine the 
inequality 
> = < Pii^o) ; 
and this may be considered for any value of Pq greater than unity. By taking Pq very 
large and k very small the inequality becomes 
(V _ ] ^ 
I > = < K. 
or 
