289 
gate guide-chords pa, v’a of a fixed guide-sphere, which 
passes through the centre a of the ellipsoid. 
We may also say, that if of a quadrilateral (aBED’) of 
which. one side (aB) is given in length and in position, the 
two diagonals (az, BD’) be equal to each other in length, and 
intersect (in D) on the surface of a given sphere (with centre 
c), of which a chord (an’) isa side of the quadrilateral adjacent 
to the given side (ax), then the other side (BE), adjacent to the 
same given side, is a (polar) chord of a given ellipsoid: of which 
last surface, the form, position, and magnitude, are thus seen 
to depend on the form, position, and magnitude, of what may, 
therefore, be called the generating triangle anc. ‘Two sides 
of this triangle, namely, Bc and ca, are perpendicular to the 
two planes of circular section; and the third side aB is per- 
pendicular to one of the two planes of circular projection of 
the ellipsoid, being the axis of revolution of a circumscribed 
circular cylinder. Many fundamental properties of the ellip- 
soid may be deduced with extreme facility, as geometrical* 
consequences of this mode of generation; for example, the 
well-known proportionality of the difference of the squares of 
the reciprocals of the semi-axes of a diametral section to the 
product of the sines of the inclinations of its plane to the two 
planes of circular section, presents itself under the form of a 
proportionality of the same difference of squares to the rec- 
tangle under the projections of the two sides Bc and ca of the 
generating triangle on the plane of the elliptic section. 
If we put the equation (35) of an ellipsoid under the form 
: T(ip + px) = K— v7 (44) 
the constant vectors . and « will be in the directions of the 
normals to the planes of circular section, and may represent 
* For the following geometrical corollary, from the construction assigned 
inbove; the author is indebted to the Rev. J.W. Stubbs, Fellow of Trinity College. 
If the auxiliary point p describe, on the sphere, a circle of which the plane is 
perpendicular to sc, the point £ on the ellipsoid will describe a spherical conic. 
