ON KOUIl.A TKKAI. TKIAXCLKS. t)27, 



Heiu-e';;'' (a- + 'S h-)- + -'Jl (/>- + :h>-y- = {^a- — h-)- - - - - (iii), 



. ;- • "- 0- 



and k\ kA--, are the roots of (X-- '^''' /.■-} - ''' ^' ( '^"' /■ _ 1) = 0. 1 



"■- "-~'^ '>'- ' (iv). 



i.r. (F'a- - 3 /•- //-) --' — (-An- /• - />-') fr = 0. | 



.-. the ceiitve of any eqxiilattral triangle iiiscfibed in the ellipse 



lies on the coaxial ellipse ^ + •' ^ = 1 - - - - (iii). 



II - 'i - 



And the radius R of the triangle is given by 



{o- — h-y 



and troni (iv . = = L =z -y L 



R 



., _ 16 {a'p + 6 '7-2) 



Also, if .r' (r/-^ — //-') E 7 (a-— 3 //- + 2,1' k ) = —~ (,r^ + 3 A-') 

 and tf{n- — 6-') E y ( — />- J- 3«- — 2 /y-'^- ) = f/ 0>' + 3^/-), 



we have — = — — and —=:-)- — 

 a a' It h 



Hence the following construction f^r inscribed equilateral 

 triangles : with any point (~'.y) on '~—^ (i/- + 3 b-)- + ^^ {h- -\- 3 n-)'- = 1 



as centre, draw the circle which passes through the point (,/, — //) 

 where / // lias the same eccentric angle on the original ellipse as 

 X y has on the inner one: the circle cuts the ellipse in 3 other 

 points which are the corners of an inscribed equilateral a with 

 x !J as centre. 



Now, for tetrahedra inscribed in ellipsoids, one of the faces must 

 be an equilateral triangle inscribed in an ellipse. Suppose the 

 direction cosines of this section to be {lin n), then the elementary 

 work on sections of an ellipsoid gives for the co-ordinates of the 

 centre Hln-, Omb-, Hue-, and for the principal radii ka, k p. where 

 it. ij are the principal radii of the ])arallel central section, given by 



I' III' II' 1 , . .,,.,,.., ,1 



1^1 1 ■•" 1 I =<'. and /•- + «-(</-/- + />-»/- + r-//-)= 1. 



//'- /- h- I- c- /■'- 



