PROFESSOR O. H. DARWIX OX ELLIPSOIDAL HAR.MOXIC AXALYSIS. 
4(;r) 
or else 
a= = + 1), c>! = i-'*(r+ 
1 — rj! 
The developments would then proceed hy ])owers of -q. 
In order to discover Avhat is the greatest value of y] w^hich must be used so as to 
comprise all ellipsoids, when we proceed from both bases of development, a com¬ 
parison must he made between this assumption and the previous one. Suppose in 
fact that 
> 3/0 1 - + /3\ 
= “ rrW 
= : 
E = 
_ 1 ) = + 1 ); 
0 
C" 
= = 
e ( r 
' + 1 )■ 
Then 
J.) 2¥I3 
h- — a-' = - 
; c' 
- E 
- 10 r = ‘ , 
1 - 0 
] - 7 ;’ 
and therefore 
j ( 
Cl 
1 
_ P? 
or 7) 
_ 1 - 3;8 
1 - 8 
1 + 7 ; ’ 
t + 0 
When -q and /3 are both equally great, they must each equal the positive root of 
1 — -I/S . 
lliis i-oot m ^^/5 — 2 or •236. 
1 -f ^ ■ 
Thus the greatest values will he 
In this case = /3' = very nearly, whereas when ^8 = /S" = 9 . Thus if the 
developments were to stop with /3^ we should double the accuracy of the result. 
However, I do not at present propose to carry out tlie process suggested. 
§ 3. "The Differential Equatiotu 
We now put uff- = D . 
rr = - yl’ 
ol + /3 
1-/3’ 
and find from the formula of § 1 , 
= - k\ c- = 0; 
l~/3/o l-)-/3'j/„ l-e/3\ oj. 
3 = — (h' — 1) iff — l)siir^. 
1 — yS cos ' 24 ) 
0 = 
. ( 1 ). 
