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).