334 
ON THE CENTRO-SURFACE OF AN ELLIPSOID. 
[520 
and we thus have 
v ' -Pff. + Q(f + £.)+3fl 
which, considering ^ as a given function of f, gives rj as a function of £. 
33. I write f = 2x, £ — £ = 2y, so that p = 2x, 5 = x 2 — y 2 . The relation between 
£, takes the form 
6 (P + Qx + Px 2 — x 3 ) — (6x + 2P) y 2 = 0, 
or, what is the same thing, 
2 - ( x + a2 ) ( x + ^ ( x + c ' 2 ). 
y x + i (a 2 + 5 2 + c 2 ) ’ 
so that taking x at pleasure and considering y as denoting this function of x, the 
values of belonging to a point on the nodal curve are |=(x + y), % 1 = (x- y); 
and the value of rj is then given as before. 
34. The form just given is analytically the most convenient, but there is some 
advantage in writing x, — y, in the place of x, y respectively; viz. we then have 
(x + a 2 V2) (x + b 2 V2) (x + c 2 V2) 
x + £ V2 (a 2 + 6 2 + c 2 ) 
1 1 
where f = -p_(x + y), £ = -= (x — y), so that if (£, £ a ) be taken as rectangular coordinates 
v2 a/ ° 
V2 
of a point in a plane, (x, y) will be the rectangular coordinates of the same point 
referred to axes inclined at angles of 45° to the first-mentioned axes respectively. 
35. The curve is a cubic curve symmetrical in regard to the axis of x, and having 
the three asymptotes, 
x = — i (a 2 + b 2 4- c 2 ) VI, y = ± jx -f |(a 2 + 6 2 + c 2 ) VI}, 
viz. these all meet in the point P the coordinates of which are 
x = — ^ (a 2 + b~ + c 2 ) V2, y = 0 : 
moreover we have y = 0 for the values x = — a? V2, — 5 2 VI, — c 2 V2, that is, the curve 
meets the axis of x in the points ri, B, G\ the order in the direction of — x being 
C, B, P, A as shown in the figure: and with these data it is easy to draw the curve: 
the portion which gives the crunodal part of the nodal curve is that extending from 
B to the points il; viz. at B we have f = £ = — 6 2 , corresponding to the umbilicar 
centre; and at il, il we have f or & = — c 2 , £ or £ = - c 2 + , corresponding to the 
/3’ 
outcrop.