130 
ON DIFFERENTIAL EQUATIONS AND UMBILICI. 
[330 
of an ellipsoid, and taking a, 0, 7 as the coordinates of the umbilicus, and 9 as the 
inclination to the axis of x of the tangent to the principal section through the 
umbilicus, then transforming to the umbilicus as origin and the new axes through 
that point, viz. the axes of x, z being the tangent and normal in the plane of ac, 
and the axis of y being at right angles to this (or in the direction of b), the equation 
becomes 
(a + x cos 6 — z sin 9) 2 y 2 (7 — x sin 6 — z cos 9) 2 _ n 
+ ¥ + c 2 - 
a“ 
or, expanding, 
a cos 6 7 sin 9 
a 2 b 2 
2z 
a sin 9 7 cos 9 
— 1 2— 
a 2 c- 
But we have 
„ /cos 2 9 sin 2 9\ y 2 /sin 2 9 cos 2 9\ 0 . ü a ( 1 1 \ A 
[a 2 c 2 ) b 2 \ a 2 c 2 J W c 2 J 
■ / a 2 — b 2 c V& 2 — c 2 
a = a a / , 7 = 
V - r? 
tan 9 = 
a? — c z V a 2 — c 2 
c Va 2 — b 2 
a Va 2 — c 2 
and thence 
. n c\/a 2 —b 2 c s\ a^b 2 —c 2 a 
Sin 9 = r—, = T7. a > cos ^ — z, /——= f7 
6 Va 2 - c 2 ~~~ “ b Va 2 - c 2 be ’ 
and substituting these values, the equation becomes 
_ b x 2 y 2 „ (a 2 + c 2 ) 6 2 — a 2 c 2 Va 2 — 6 2 V6 2 — c 2 A 
— 2^ 1- ^—i- ——[- z 2 -— ! —4—^ 1- 2 r- zx = 0, 
on 7*2 7*2 q2Jj2q2 hftp.n. 
ca b 2 b 
or, what is the same thing, 
ca . , „ 1 /—— r „ /fr a 2 b 2 + c 2 b 2 — a 2 b 2 
z = . |•(¿c 2 + y 2 ) + ^ va 2 — 6- v6 2 — c 2 H 77TT z*; 
2b 3 ac 
whence approximately 
ca 
* = + V% 
and thence to the third order in x, y, 
z = • 2 (^ 2 + V 2 ) + ^ ^ a 2 -b 2 a 2 -c 2 x (x 2 + y 2 ), 
which is of the form in question. 
5, Downing Terrace, Cambridge, November 2, 1863.