PROFESSOR G. H. DARWIN ON ELLIPSOIDAL HARMONIC .ANALYSTS. 
Hence for the coefficient of the first series we have 
Bnt II (— 
[n (n-i)]~;r : 
2/1! 
n ! 
t n II (-i) 
n'(0)-i^f|^=iog.4» 
11 (n) n {n — l) = n ! n — l ! 
2^-1 
Therefore 
27i! n(?i —1 + 6)11(?i, + e) /i!?/, —Ilf / 1,^.^^! 1 \ 
^^2++ [n(.-i+e,r =2-^-|l + ^(2log4--+2^--4^ 
2^-1/ ■ 
This is true from n=co to 1, but in the case of n = 0 we have 
so that in that case 
n (-!+£)=+(£)=i+n'(o), 
n(-l + e)n(e> i,„, , 
=:+3 i"g 
[n(-i + 6ff e 
Now consider tlie coefficient of the second series. 
We have 
and since 
= + log,/c), 
77 
n i—x) n {x—i)= r 
' ^ ' SlU TTX 
n ( —n—1—e) II (?i + €) = - 
77 
(-) 
n +I 
■TT, 
sill (n + 1 + e) 77 6 
Therefore the coefficient of the second series is (—) 
[n(-i)]-=>T. 
2// ! /c"“ 
hf log 
D = 
2n ! 
/ 111 
l + eh log 4--+2.5'j-4.S'-— 
22M«0- e 
^ih i’ l—n — e, K~) 
2«,i 1 
_|_( — Y+^——— _ ^X-|-2e log k) F (w+-|-d-e, n + i + e, r<,+ l-f-e, k~). 
The case of n=0 is an exception, for the coefficient of the first F has the part 
Inside [ ] replaced by - (l + 2e log 4). 
* Proved by differentiating the known formula II(j!:- 1) II (.r - 4) = H (2 x - 1) 
.; = 4. 
(Itt)! 
44 : 
and putting 
