PKOFESSOE C4. H. DARWIN ON ELLIPSOIDAL HARMONIC ANALYSTS. 
471 
We assume then the following forms for the functions :— 
For EEC, OEC, OOS, EOS 
For EES, OES, OOC, EOC 
P' = nh/,,p+ + N;8V,^2 „p^+^"} 
In these series a proceeds by intervals of one at a time, beginning from a lower 
limit of unity. In both forms the upper limit of the first N is or ^ ( 5 — 1) according 
as s is even or odd ; and the upper limit of the second N is F (f — s) or ^ (f — 6- — 1) 
according as i and s agree or do not agree in evenness or in oddness. 
The factor O contains (w — 1)- in the denominator, but does not become 
infinite when v = because when .s- is not zero P'^ is divisible by {y — 1)^' and 
we shall see that is zero."^ When s is zero there is no function of the P type. 
It may be noted that the limits of the series are such that neither q nor q'. can ever 
have a negative suffix. 
We shall ultimately make qg and q'g equal to unity, and this will he justifiable 
because there must be one arbitrary constant. 
We have now to consider the forms of the cosine and sine functions. They may 
be derived at once from the preceding results, for we have only to read (w — 1 )-^ 
0 1 + /d - 
as cos tcj) where t is even ; (h’ — 1)“ as sin </>, — .-) as cos f/), and v as 
1 fJ 
(1 — cos 2(f>)K 
'I'he factor (1 — /3cos '2(f)f will occur frequently, and I write 
q> (<^) = (1 — 13 cos 2<^)", 
and as elsewhere I commonly write to denote di 
The folio wins: are the results :— 
Type O or EEC ; 
€ 
s _ 
y/ COS t(f). 
Type AB or EES ; = Syt sin t(j). 
It is clear that we may equally well regard the lower limit in the latter as zero. 
Type A, or (i)OC ; each term is of type cos {t — 1) (f) cos cf) or cos (^ — 2) ^ + cos t cf). 
i 
Hence (IT'* = lyt cos f(f). 
* This also follows from the fact that the series for P*’ begins with Pa., (v- - 1) iu the case of EES, 
and with 9.ol^v - 1) in the case of OES. Thus in the former case there is no term Pao and in the 
latter no term Pa;v 
