289 



gate guide-chords da, d'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 (ae, 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 (ad') is a side of the quadrilateral adjacent 

 to the given side (ab), 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 abc. 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 diflference 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 {cp + pk) = ^- i\ (44) 



the constant vectors i and k 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 

 above, the author is indebted to the Rev.J.W. Stubbs, Fellow of Trinity College. 

 If the auxiliary point d describe, on the sphere, a circle of which the plane is 

 perpendicular to bc, the point e on the ellipsoid will describe a spherical conic. 



