344 
ON THE CENTRO-SURFACE OF AN ELLIPSOID. 
49. We have then 
2 o 7 
a¥ = 
/3 
1 + r. 1 + Í, 
7 s A + r + 3q + 3g(r + g)\ 
/3 V 7 7 2 / ’ 
= -t( i+!A3î_6 ÿ 
/3 V 7‘ 7 
“ £ 1 1 + q* 
r r 
4 (7 — a) 6a 
_ _ 7_ 
/3 
1 + q 2 
7“a 7-a 
2 (/3-7)1. 
7 2 a 
• > a? / v\ [ v\^ 
and in the same way from cV = — ^ ^1 — -J ^1 — -J , we have 
moreover we have at once 
/3 
6y = 
7 oc 
gV _ 3ç 4 
7a 7a 
Hence, writing x + Bx, 0 + By, z + 82 for x, y, z, we find 
._!* 2 (^-7) 
2 ' 7 2 a 
-2 (a-13) 9 
OZ —iz . . 0-, 
2 7a 2 1 
or, what is the same thing, 
a, : & = : ± * JI : , 
7 2 a 6 V 7a 7a- 
where ¿r, 0 denote the values at the umbilicar centre. 
a J x-‘ = 
Nodal curve in vicinity of real outcrop, viz. 
z — 0. Art. Nos. 50 to 52. 
/33 (7 - a y _ «3 ( a _ 7 )3 
7 (a — /3) 3 ’ ^ 7(«-/S) 3 ’ 
2 4- *7 
ga/3X2 
. a 
50. Write