1896.] the plane sections of surfaces. 199 



Let (X,, n, v) be the direction-cosines of the normal to the plane 

 of the section. 



The equation is 

 a (£ + ku + Itfif + b(r)+ n\u + m 2 v) 2 + c (£ + rtjU + n 2 v) 2 = 1. 



As £, 7}, f do not enter into the highest terms, all parallel 

 sections must be similar and similarly situated conies. 

 Suppose the origin to be the centre ; then 



af& + brjnh + c£«i = 0, 



ajjla + bwnu + c£w 2 = 0, 



so that of = ^ = c|-_ 



Thus the locus of the centre of a section parallel to a given 

 one is a straight line through the centre of the surface. 



If the axes of the section are the axes of u and v, we have 



al-^L + bmjiu + cn^h = 0, 



lj 2 + wixWla + n{ii 2 = 0, 



lj,. 2 m{iiu njn s 



or 



— c c — a a — b 



Substitute for (L, m 2 , n 2 ) in the equation \L + finu + vn 2 = 0, 

 and we find 



^ b — c c — a a — b 

 \-.— +p. + v — = 



as the equation that determines the directions of the axes of the 

 section in terms of X, p,, v. 



Or we may eliminate (l 2 , m 2 , n 2 ) by means of multipliers and 

 write 



ah = pli + a\, bm^ = pnii + apu, cih = pn x + <rv. 



Thus p = al-c + 6wj a + cn^, 



so that (1 — af 2 — brf — c£-)/p is the square of the semi-axis in the 

 direction (l 1} m 1} n-^) ; l lt m,, w x are proportional to 



\ p. v 



a — p' b — p' c — p' 



j ^- 2 A*- 2 ^ 2 n 

 and 1- jJ- 1 = 0. 



a — p b — p c — p 

 If the section is a circle we have 



a (Ij 2 — l 2 -) + b {m x - — m£) + c (V — n?) = 0, 

 aljj 2 + bmiin.2 + engirt = 0. 



