520] 
ON THE CENTRO-SURFACE OF AN ELLIPSOID. 
341 
45. I will show that these values of £, give the foregoing values rj = = — a 2 . 
We have 
1 : v~Vi • V + Vi = P&i + Q + &) + 322 
this is 
or 
(£ + ?){ ^-(p-^ + ft-KP + i + fO}, 
1 : 77 — 771 : r) + rj x = a 2 (b 2 + c 2 ) : 0 (£ x — f) : - 2a 2 a 2 (b 2 + c 2 ), 
V — Vi — 0, V + % = — 2a 2 ; that is, rj = r] 1 = — a 2 . 
46. For the real umbilicar centres and outcrops we have 
I. 
I = — b 2 , v = — b 2 , & =? — b 2 , Vi = — b 2 . 
X’ = -a? 
V = 0, 
7.) 9 ® 
c p, 
X 2 = -a 2 
Fi = 0, 
/3’ 
7 2 o 2 — 
c /3- 
a-«- = — 
/3’ 
II. £ = -c 2 , 
y = 0, \ (real umbilicar centre). 
a 3 
= — 
or 
/3’ 
7J = — C 2 
9 a/3 
(a - /3) :i 
(a-/3) 3 
2J _ 3a/3 
f, = -0-+^, * = -*■, 
V.__ 2^ (7 ~ a ) 3 
7(«-/3) 3 ’ 
■p- fr,«(/3-7) 3 
7 (a-/3) 3> 
£=0, 
ellipse concomitant. 
a y- ^ (7 ~ a ) 8 \ 
- 7 (a-/3) 3 
&Y = -- (/3 ~^ 
^ 7 («-/3) s 
z = 0. 
X 2 = — a- 
2 /3 7 ~ « 
y a-/3’ 
F 2__ 
1_ 7 «-/3’ 
^ = 0, 
ellipse sequential. 
(real outcrop).