498 PROFESSOK G. H. DARWIN OX ELLIPSOIDAL HARMONIC ANALYSIS. 
In fact, whilst we have as before, when i and s are both odd or both even, 
1 
!Zi-2 
^2 _ -f /3 c7 ’ 
we also have 
2a,- 
Y/ + 2 
1 
a + 2)2 - s2 + - i/32{h i +A] {i,i + 3} 
' 2i + 2 
and similarly a fraction for ^ and so forth. 
2 ' + 2 
It follows therefore that v>diile the qs, with suffixes less than or equal to i depend 
on finite continued fractions, those with suffixes greater than i depend on infinite 
continued fractions. 
It thus appears that 'wffiile the first series in the expression for (Q;® or for Q/ has 
limits 1 to or (s — 1), as before, the limits of the second series are 1 to qc . 
Thus we have found an expansion for this class of functions in powers of /3. 
In the limited case in which the coefficients have been actuallv evaluated, namelv, 
Achere the development is only carried as far as the squares of yS, we have 
5's -4- 2 - 
/ 
2 s +[2 - 
8 (s + 1) ’ 
g + 2 
8.s(.s + 1) ’ 
5^5 T 4 
? 5 + 4 — 
1 
128 (s + 1)(S + 2) ’ 
_ .9+4: _ 
128s (s + l)(.s + 2) • 
These coefficients do not vanish when 5 + 2 or 5 -j- 4 are greater than f, and this 
confirms the conclusion already arrived at. 
In spherical harmonic analysis there is no occasion to consider the value of Q.® when 
s is greater than i, and the values are therefore not familiar. I will therefore now 
determine them. 
Q,= 
It is knownthat 
2}a \f 
2i + 1\ 
1 y +_2 ! 
i + 1 “r o 1 I 
{2i + 3) V 
+ 
■i + 4 ! 
22 . 2 ! ^ I (2i + 3) {2i + 5) 
« + -5 i 
Therefore difterentiating 
+ 
,2; + 2 ' 
And 
~dv 
Q, == (-)‘ + ^2' . i\ 
2* i I 
zii( —)‘ + i-^^ 
2* + h 4 + 1 I 
•i + 1 1 (i -f 1) ( i + 2) 1 
1! 2~! ■ 4--' + *^ 
Q4= (-y 
(v^ - 1)‘ 
i -f 2 
V. 
* Bryax, ‘ Camb. Phil. Soc. Proc.,’ vol. 6, 1888, p. 293. 
