440 
Proceedings of the Royal Irish Academy. 
XXXVII. 
O^^ THE PLAXE CIECULAE SECTIOXS OF THE SUREACES 
OE THE SECOXD OEDER. By the Eight Eev. Charles 
GmvES, D.D., Bishop of Limerick. 
[Read January 13, 1890.] 
As a contribution to the theory of the surfaces of the second order, 
the Bishop of Limerick desires to state the general Proposition : — 
That their umhilics and the circumferences of their plane circular sections 
nre to the surfaces ivhat their foci and focal circles are to the plane conies. 
The identity of the relation is proved by the existence of the 
same equation between the radii of the circles and the distances of 
their centres from the centre of figure. 
In order to avoid multiplicity of cases, let us deal with the 
ellipsoid. We find that 
d^ r^_ 
P ~ ' 
where u is an umbilicar semidiameter on which the centres of one 
series of circular sections lie ; d is the distance of the centre of one 
of those circles from the centre of the ellipsoid ; r is its radius, and I 
the mean semiaxis of the ellipsoid 
w2 ^2 
a^ c^ 
The following method of determining the normal at the point 
(a, y8, y) on that ellipsoid is closely related with the theorem just 
stated, and presents a simple method of conceiving the genesis of the 
circular sections. Through the point (a, o, P) in the plane of the ellipse 
(a, c) draw two chords parallel to the semidiameters of that ellipse which 
are equal to h, the mean axis of the ellipsoid^ the centre and radius of the 
circle which passes through their four extremities {for they will lie in a 
circle) will he the centre and radius of the sphere which intersects the 
ellipsoid in the two circular sections passing through the point {a, y); 
and the radius drawn to that point will he the normal at it. And further 
— As a sphere inscribed in a right cone^ ivhich stands on an elliptic base, 
intersects that ellipse in a focal circle, so a sphere inscribed in a right 
cone, ivhich envelopes an ellipsoid, intersects that surface in a pair of 
circula/r sections. These planes pass through the line common to the 
planes of contact of the right cone ivith the sphere and the ellipsoid, 
forming with them a harmonic cahier. 
