520] ON THE CENTRO-SURFACE OF AN ELLIPSOID, 
343 
whence multiplying by 7, a, and adding, 
(7 + a) jr + 3g + ^ (r 3 + 3g 3 ) j = (7 + a) jn + % + 3“ ( r i + 3i Zi 3 )| , 
which, neglecting terms of the third order, is 
r + Sq = r 1 + 3 g x . 
Subtracting the two equations we have 
[I+5 (*■ ■+ % 2 )+* (? - $ (»*+ s 9*>=i (5+5) w+«+* (? - A) ( 
viz. this is 
r 2 + 3 q 2 +1 -—- (r® + 3 q 3 ) = + 3 q* +§ ——- (r x 3 + Sq^), 
73c 70c 
7 — ct 
or, what is the same thing, 
r 2 — r{- + 3(g 2 — qi) + § 7 [r‘ — r a 3 + 3 (<f — q 3 )) = 0, 
which, putting therein r — r 1 = — S(q — q^), 
is 
-r-r a + o + g 1 + §-—- (- r 2 -rr x -r x 2 + g 2 + gg : + g, 2 ) = 0, 
7 ct 
say this is 
combining herewith 
we have 
and 
where 
-r-r 1 + g + g 1 +2A=0; 
r - n + 3g - 3g x = 0, 
r + q — 2g x — A = 0, 
rj — 2g + gi - A = 0, 
A =1 (- r 2 — rr x - rf + g 2 + ggj + g^). 
7® 
But substituting herein the values r = — q 4- 2g x , r x = 2g — g l5 this becomes 
and then 
that is, 
A = i (- 2g 2 + 4gg x - 2g 1 2 ), = - f g 2 , 
r = - g + 2g 2 + A, 
r + 3g = 2(g + g x ) +A, =-* ^g 2 . 
n 3 + 3g x 3 ),