534 
PEOFESSOR G. H. DAEWIN ON ELLIPSOIDAL HARMONIC ANALYSIS. 
or, the usual notation for hypergeometric series, 
D = 
in ; 
TT 
22y„, !)2 
F{1, i u+1, 
This series is of no service, since it proceeds hy powers of which in our case is 
nearly unity. It is required then to transform the series into one proceeding hy 
powers of k-. 
It is known that, if 1, 
7-,/ 7 ,.n n(c—a —5 —Dllfc —1) , 7 1 , ,7 o\ 
F{a, h, c, y — t. _i\ ^("> !+«+& —c, k') 
+ k 
IT (c —« —1) n (c —5 —1) 
„-Mn(fl + 5-c-i)n(c-i) 
2(c- 
n(rt-i)n (5-1) 
F{c — a, c — h, c —a —«-).* 
If we apply this theorem with a — b — c = n + 1, the first F becomes 
F{^, 1 — n, K'), whose rdh and all subsequent terms involve zero factors in the 
denominators. Also the coefficient of the second F involves n( —n — 1), which 
has an infinite factor. Hence the formula leads to an indeterminate result. Let us 
therefore put c = 7? + 1 + e, and proceed to the limit when e = 0. 
We have then 
D = Limit rr — 
%i\ f 11 (A—1 + e) 11 (71+ e) 
92« (77 qa 
[n {n~i + e)f 
F{h l—n — e, K') 
^ y) ” F (n+i+e. «+i+e. n+l + e, K-)\. 
Now n(e)=i+en'(0), n(-i+€)=n(-|) (i+€k7rtl)- 
* 
Therefore, when e is very small. 
II (a — 1 + e) 
= + ... +i+i)+£n'(0) 
=i+qn' (o)+hLh 
t nj 
1\ 
ri(a-iAe) . . 1 1 , 
n(a-4) 
= 1+6 
+i+l) + e 
Elzi) 
n(-4) 
1 
2^-1 
*. I have to thank Mr. Hoesox for giving me this formula, and for showing me the procedure whereby 
it can be made effective. 
