348 
ON THE CENTRO-SURFACE OF AN ELLIPSOID. 
[520 
and writing 6' 2 + </> 2 = X, 36' 2 — (f> 2 = Y, the first of these is 
( 7 - a) 2 _ - 9 (X + Y) (X + 1Y 
ya 
(F+3) {3 (X 2 — 1) + F + 3} ’ 
which regarding X, Y as the coordinates of a point in a plane is a cubic curve having 
the point X+1 = 0, F+ 3 = 0 as a node: hence writing F+ 3 = Sa (X + 1), X and Y 
will be each of them a rational function of <r. The second equation is 
6flp(X + l) . Of- «)<r = (7-_?)?• 
— — 7 cl, rnar is, jj 20 > a / v '~,—r? ’ 
F+3 
Y X + Y 
and we have also 
2e = \fX+Y, 2(f) = s/SX — F ; 
the equations thus become 
( 7 - a )<r(X+j0j 1 + i/ y 
V 
= -¥ + 
8 
SX- F) 3 
X + Y 
which are better written in the form 
£ = -& 2 -£(7-a)c--jl -\J 
-3 X + Y 
X+ Y 
?/ = _ 6 2 + H7 _ a ) <7 (Z+F){l + / V /-|^| S , 
where X, Y are given functions of cr. We in fact thus obtain an analytical expression 
of the nodal curve, quite independent of the considerations as to real and imaginary 
which suggested the process: the foregoing values substituted for £, r/ will give 
— fiya 2 ® 2 , &c. equal to rational functions of cr, so that taking for fj, % the same expressions, 
only changing therein the sign of the radical ~~x +Y~ ’ ^ ese va ^ ues > Vi gi ye 
the very same values of — ftya-x 2 , &c., or the values of £, rj, rj l satisfy the conditions 
(a 2 + £) 3 (a 2 + v) = O 2 + £i) 8 (a 2 + Vi)> &c. 
for a point on the nodal curve. 
55. To complete the investigation, writing as above F+ 3 = 3<r(X +1), we obtain 
(7 — a) 2 _ (3cr + 1) X + 3cr — 3 
— 7a cr (X + cr — 1) ’ 
nr putting for a moment 
(7 — *)*<r = K 
— 7 a.