488 
rROFESSOE G. II. DARWIX OX ELLIPSOIDAL HARMOXIC AXALYSIS. 
§ 7. Rigorous determination of the Functions of the second degree. 
If a numerical value be attributed to /3 it is obviously j)ossible to obtain the 
rigorous expressions for the several functions. Thus, if /3 were \ we could determine 
the harmonics of the ellipsoids of the class m ^ {cd + F). But I do not think it is 
possiljle to obtain rigorous results in algebraic form when i is greater than 3. In 
order, liowever, to show how our formulge lead to the required result I will determine 
the live functions corresponding to i = 2, hut I will not work out the case of i = 3, 
althougli it is easy to do so. 
When s = 0 
/3(t 
i/3d2, 1} {2, 2} 
4 + /So- 
12/3- 
4 + /3a 
Tlierefore /3cr = — 2 + 2(1 + 3^")^, or writing = 1 -f- for brevity, 
(Bo-— — I). Then putting (2o = B and remembering that the value of is 
twice tliat given by the general formula, 
1 - I 
~ 4 + 13a ~ • 
Therefore ^3 = P 3 + ^ P/. (If), 
wliere P., = fa- — 1 , Py = 3 (a- — 1 ). 
The coefficient ol the cosine function is given by 
Therefore 
Pz — ■“ (2) l]W-2 — ~ 
jf _ 2 
Co = 1 — —cos 2<^.(18). 
5 =1, cosines ; EGO type. 
The continued fraction has 2 ]{?, 3} in the numerator, and vanishes because 
[2, 3} = 0. Therefore 
'fherefore 
(Ba -f \IBi {i I) = 0, wliere i — 2 . 
But the coefficient is independent of fB, for 
Therefore 
= n [^'/Po'], and qf = 1 . 
P„i = \/- 7 Po^ = 3a( a- 
V _ 1 ~ \ 
1 + F d 
1 - 
• (19)- 
Clearly 
tj 
Co^ = cos (j}{\ — (B cos 2(^,f 
• ( 20 ). 
