Manchester Memoirs, Vol. Ixiv (1920J, No. 1 3 



the right hand side of this equation so that e disappears. Now in the 

 ellipse r = a — e. CN or e. CN—d — r. Now take a point N on CA 

 such that CN' = a - r = e. CN, and let jj! be the eccentric angle of the 

 corresponding point P' on the ellipse. Then clearly // = /3. Hence 

 Adams' proof leads at once to a simple construction for a and jo. To 

 construct (3 + a and (3 - a (or ty x and <j> 2 ) let Q correspond on the 

 auxiliary circle to P' on the ellipse, and take two points Q\ and Q' 2 on 

 opposite sides of Q and such that | Q' I CQ'= \ Q CQ 2 = \ \ Q X CQ 2 . 

 Then the angles ACQ' Z and ACQ 2 {jjl \ and \x 2 say) are equal to <p x and <j> 2 . 



These results suggest that an independent proof of the theorem may 

 be given, based on geometrical considerations. Let P, P\, P / 2 be 

 constructed as above. Then with the same notation as before we have 

 r T + r 2 = ia — ae cos jjl 1 — at cos jjl 2 . 



= 2a-2 ae cos \ (^ + /j, 2 ) cos ^ (fi z - fj. 2 ) 



= 2a — 2 CN' cos \ (fji'j — fi 2 ) 



= 2a — 2 a cos \ {fi\ + fjL 2 ) cos I (fi\ — // 2 ) 



— 2a - a cos u\ — a cos /j. ' 2 



= 2a-CN\- CN' 2 , 



= AN' T + AN' 2 . . . • . . . ' . . (9). 



Again k 2 = 4a 2 sin 2 J (fj 1 - /z 2 )[i - e 2 cos 2 J (//, + /x 2 )] 

 = 4« 2 sin 2 i (jii x -fi 2 ) sin 2 J (//x + z/*) 

 .'. £ = 20 sin \ dj! x - fj! 2 ) sin J (//i + /*'») 

 = « cos /j! 2 — a cos fj! x 

 = N\N' 2 (10). 



Hence, from (9) and (10), 



f / =sinK' I = sin| ( ^ I , ^ / — = sin \ «' 2 = sin £<f> 2 . 



J a si a 



Also «/ = (^x - ja 2 ) — <?(sin Ml — sin ^ 



= (c/)i — <^ 2 ) - 2 <? sin ^ - ^3) cos J (/xi + /z 2 ) 

 = ((/)i —<t> 2 ) — 2 sin J («£, - fe) cos J (</>! + </> 2 ) 

 = (cf>i — sin </,,) — (</> 2 -sin <£ 2 ). 



Although very different in form from the proof given by Adams the 

 preceding proof is not very different in substance, depending, as it does, 

 on expressing r + r, k and nt in terms of ^ - ^ 2 and e cos \ (/xi + ^ 2 )- 

 It is, however, of some interest as placing the matter on a definitely 

 geometrical basis and for the immediate purpose for which it is here 

 used appears to be superior to the geometrical proofs hitherto given 

 c.f. C. Taylor's " Ancient and Modern Geometry of Conies," page 241. 



