180 



This is the tangential equation of the " surface of centres of curva- 

 ture" or as it may for brevity be called, the surface of centres. 



This surface in general consists of two sheets, one generated by 

 one centre of curvature, the second sheet by the other centre. Let 

 a perpendicular P on a tangent plane to the surface of centres make 

 the angles X, /*, v with the axes of coordinates, then P=cosX, 

 Pv=coSju, P=cos v, and the last equation may be written 



[~COS 2 X COS 2 U COS 2 V~1/ o 9v , 79 9.9 9 TV>\ /^\ 



1 = ^- + ^ 4. 2 cos 2 \ + 6 2 cos> + c 2 cos 2 i> P 2 ). (7) 



Now the first member of this equation represents -, the inverse 



K 



semidiameter squared of the original ellipsoid, making the 'angles 

 X, p, v with the axes, and a 2 cos 2 X + 6 2 cos 2 /* + c 2 cos 2 v=V? is the 

 square of the perpendicular on a tangent plane to the ellipsoid parallel 

 to the tangent plane to the surface of centres. Hence 



P 2 =P 1 2 -R 2 ........ (8) 



Whence we have this remarkable property of the surface of centres : 



Any two parallel tangent planes being' drawn to the surface of 

 centres and to the ellipsoid, the difference of the squares of the co- 

 incident perpendiculars let fall upon them from the centre is always 

 equal to the square of the coinciding semidiameter of the ellipsoid. 



We may reduce the original equation (6) to the form 



(9) 



By giving to a set of constant values, we might determine the tan- 

 gential equations of the sections made in the plane of xy by the cone 

 whose vertex is in the axis of z, and which envelopes the surface of 

 centres. t 



But it will be better to determine the sections of the surface made 

 by the principal planes, and this may be effected by putting , v, 

 successively equal to oo and 0. Hence we shall have in the planes 

 of yz, xz, and xy, the sections whose tangential equations are 



I in the plane of zy, 



I in the plane of xz, 



a" c 2 ) 2 6 2 2 + (6 2 c 2 ) 2 V= 2 6 2 . 



' in the plane of xy. 



