PROFESSOR G. H. DARWIN ON ELLIPSOIDAL HARMONIC ANALYSIS. 489 
6 ' = 1, sine ; EOS type. 
The continued fraction again vanishes and cr = 3, but it is not needed to express 
the functions. Putting p,' = 1, 
Pd = - 1)^.(21), 
S_;^ = sin<^(l — /3 cos 2(^)^ . ..(22). 
.9=2, cosine ; EEC tyj^e. 
The second continued fraction vanishes because it contains {2, 3} in the numerator. 
The equation is then 
_ -1/3T2, 1} {2, 2} _ -^2/3) 
4 — ^cr 4 — /3a 
Therefore /3cr =2 — 2(1 + 3^')^‘ = 2(1 — B). Then putting q., = 1, 
, {2,21 {2, 11 - 2{B - 1) 
-L - /3a + 
Therefore 
~ p, + p,.-. 
. . . (23), 
where 
P, =: 1+ - 1, 
1+ = 3 {F - 1). 
Then 
Pu = - 19 V A = - ko> 
J/ 
^ + COS 2(f) . 
. . . (24). 
5 =2, sine ; EES type. 
Both fractions disapper.r and cr vanishes, but is not needed for determining the 
functions. Noting that q,/ = 0, and q.,' = 1, 
p,= = nfe'P/] = 3 (.5 - i tdf (- - 4 • 
I3j 
= sin 2(f) 
(25) . 
(26) . 
We can write down the functions of p Iw .symmetry, and the products of the three 
functions give rigorously the five solid liarmonic sohitions of Laplace’s equation of 
the second degree. As I have remarked above, the seven harmonics of the third 
degree may be obtained rigorously by a parallel process. 
§ 8. Aqqjvoximate Form of the Functions. 
It is clear that the first approximation to )8cr is zero, and that the second approxi¬ 
mation, in the general case, is ' ■ 
VOL. CXCVIT. —A. 3 R 
