[520 
520] 
ON THE CENTRO-SURFACE OF AN ELLIPSOID. 
345 
C. VIII. 
44 
3 aß 
and first for the relation between q and ft, writing for a moment —q l — Q l , and 
therefore = — c 2 + Q 1 , the equation of correspondence gives 
- Saß (q + Qß + (q- + 4 qQ l + Q 2 ) (a-ß) + SqQ 1 (q + Qß = 0, 
which, putting for Qj its value, is 
Saß \ 
a — ß 
— Saß ( q 4- ft ■+■ 
a — ßt 
+ (« - ß) [f + *qqi + qß + (*q + 2?0 + j— 3 
ß (« - ß)v 
that is, 
+ s q {q(q + qO + (q + ^qi)^r ß + ^^ = o-, 
- Saß (q + ft) 
+ Saß (4q + 2ft) + (a - ß) (q 2 + 4qq 1 + qß) 
+ (J-ßf q + ^~'ß & + 2qqi ^ + Sqqi ^ + = °> 
or, what is the same thing, 
[ 9aß+ iß§ß q+3ttßil 
, 9 fa 2 + 7a/3 + /3 2 , a? + 16aß + ß\ 9/ m ) 
+ q —Tr—*— + qsi—-—a— + ?i («- ß) 
a — ß 
+ Sqqi (q + qi) = °> 
«-/3 
or, for small values, 
Saß \ , . 3il 
(3 + (a-^jg + gi = 0 > that 1S > (a-ßf q + qi = 0 ‘ 
51. Moreover, from the equation (c 2 + £) 3 (c 2 + ??) = (c 2 + gß 3 (c 2 4- t/j), we have 
9a/3fl _ / 3cc/3 Y £ that is 0 = i— o 3 
1 (a-ßf \a-ß) ' 15 t t ^ ¿tfß-’ 9 -' 
or, since ^ and ft are of the same order, is of the order ft 3 . Hence, starting from 
the equations - ß<ya?x 2 = (a 2 4- fi) 3 (a 2 4- yß &c., the terms of x, y arising from the 
variation of y x are indefinitely small in regard to those arising from the variation 
pf £; and we have 
Ihx 
3ft 
-3ft 
(a-/3) 
X 
-*+**’ 
a — /3 
to 
-T 
l 
» 
%_ 
3ft 
3ft 
a — ß 
y 
3a/3 ’ 
a + 0 
a — ß 
» 
l