ON EQUILATERAL TRIANGLES INSCRIBED IN ELLIP- 

 SES AND REGULAR TETRAHEDRA INSCRIBED IN 

 ELLIPSOIDS. 



By Prof. W. N. Rose ve are, M.A. 



If 1 2 3 is an equilateral triangle inscribed in tlie ellipse 



+ -r = L 



we have (,/j — '•:..)-+ (//, — ^2)'= ('i — ''3)"+ (.'/i — //:;)-• 



... (.,., _ ,,..) (9 ,, _ ,, _ ,3) + (^^ _ ^.) (2y, - y, - //,) = 0. 



Now (.., - ./■,) ''■' + 'V (//., - y.^ -'-'^p = 0. 

 and if ./ , + ./o+ ':; = 3t, //, -f //._, -|-//., = 3(7, (7 y) is the centre of the a, 



1 3(7 //l , _, 37 .1 y _ 



and J ^y^ ^r^- ,)= ^^._, O/i-//). 



... J.,.,(„2_ft2)+ j.(3io_,,2)| |_,/^(„2_/,2)_ -(3,,.^_/,2)| 



__ _,_3r/'-3&^ + 6a-62] 

 - ' •'' / _10^/-/y-' + 3rt*4-3JM 

 = — Aa~h-~ T/. 



(i) 



.-. /J (rr' — />-) = 7 (r/- — 3 //-' + 2<r k) 



(9 / 1' 

 — h- 4- 3^/- — j, 



with similar equations for ( '._) .v^) and (./..//..). 

 .-., using ''%-^^'= 1, 



■':\ („-'_ 3/;-+ 2r/- A-)-+ K r^'' + •^"■-' - ^f 'V= ("■-' - ''-)■- (ii)- 



rr h- \ k J 



This is an equation of the fourth degree for /, in terms of 'ijj-, 

 and the ])i'opcrties (>f the roots give (the loots lieing k^ k.,k»k) 



-k + k =-"''^,^''\-.\ + ] =-'''+,^"\k,k.,k,k ^'''^\ 

 a- k k b- (t^ .f' 



Now fiom (i). since .? , + r._, -|- ' :; = 37 and //, -f //o + y., = 3.y, 



37 {fi- — />■-) = 7 (3<'/-— 9 //-+ 2a- i: /•) 



;nid 3/7 (n- - //-') = /7 (- 3 /y-' + 9 tt" - 2 A- i: ^ ) 



•. /■ = — 1 , and /•, /■., /• 



l>'!i' 



