464 
PROFESSOR O. H. DARWIN ON ELLIPSOIDAL HARMONIC ANALYSIS. 
Let the axes of the fundamental ellipsoid of reference be 
to 1 + 
1 - /3 
The ellipsoid defined by v has its tliree axes a, h, c given by 
a 
o 70/ 0 
= /c-' V" — 
1 + /3 
1 - /3/ 
, 6^ = It? (m — 1), c® = ]??’■', (a < 6 < c). 
Tins mode of defining the axis is such as to indicate the relationship to the 
prolate ellipsoid « = 6 < c. But another hypothesis may he made which will bring 
the axes into relationship with those of the oblate ellipsoid « = 6 > c ; for if we 
take a new I', numerically equal to the old one but imaginary, and rej^lace w by — 
we have 
o 
a- 
1 + / 3 \ 
1 
/3-^ = + 1), 
{a>b> c). 
If /3 be made to range from 0 to co , all possible ellipsoids are comprised in either 
of these types. It will, however, now be shown that, by a proper choice of type, all 
ellipsoids may be included with the I'ange of ^ from 0 to ^ . 
Let us suppose the axes to be expressed in three forms, as follows :— 
(1-) (2-) 
52 = _ 1) = q_ X) 
(3.) 
1 + /SnX 
— J?v^ 
= - 1 ). 
Then "we have 
Therefore 
And 
2/,:2/3 
1 - /3 “ 
7 0 1 + /So 
- “ 1 _ 
- 1? = 
1? — 
2/.’dA 
_ 2 /.d^o 
1 - ySj 
h'^ - or 
2 /? 
L - /3i 
1 + /So 
? - V ~ 
1 - ^ 
"■ 2 ^, 
~ ~ 2^0 ’ 
(j- — rd 
1 + 
I + /?! 
1 - /So 
0 70 * 
6 "" — 
1 - /3 “ 
2 A 
~ ■■ 2/3o 
Ir - a- 
/3 = 
1 
1 S- /So 
2 d - fl2 - d ~ 
1 + 3^1 
~ 1 - ;3;S 
