Manchester Memoirs, Vol. Ixiv (1920), 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=a-r. Now take a point N' on CA 

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

 corresponding point P' on the ellipse. Then clearly /*' = j3. Hence 

 Adams' proof leads at once to a simple construction for a and /3. To 

 construct /3 + a and }3 - a (or <f> r and <f> 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\CQ'= \ QCQ 2 = \ \ Q Z CQ 2 . 

 Then the angles ACQ\ and ACQ 2 (/A and fx 2 say) are equal to <f> z and 2 . 



These results suggest that an independent proof of the theorem may 

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

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

 r, + r 2 = 2a - ae cos li t - at cos ll 2 . 



= 2(2-2 at cos i (^ +/x 2 ) cos 5 (j*, - yu 2 ) 



= 2rt - 2 CA"' cos J (// x - ^u' 2 ) 



= 2a — 2 a cos J (fi' 1 + p! 2 ) cos A (/i t — /i 9 ) 



= 2a — a cos ^u'j - a cos yu' 2 



= 2a-CN' I - CN\ 



= AN\ + AN' 2 (9). 



Again fr = 4 a* sin 2 \ ( Hl - fx 2 )[i - 1* cos 2 £ (/*! + /**)] 

 = 4« 2 sin 2 i (/»'x -/*',) sin 2 J ( / x / I + / x' 2 ) 

 .*. ^ = 2fl sin i ( /X ' I - M ' 2 ) sin J (j/, + ,/,) 

 = fl cos ^'2 - a cos ^'1 

 = N\N\ (10). 



Hence, from (9) and (10), 



2 / =sin I M ' I = sin J <£„ ^ / 2 - — = sin \^' 2 = sin ^ 2 . 



Also «/=(/*, - /jt2 ) -<?(sin yui — sin M2 ) 



= Ui — ^ 2 )-2 t sin Kjfa-jte) cos J (/xx + pt.) 

 = (</>i - </> 2 ) — 2 sin J (<£ x - </> 2 ) cos J (<£ x + </> 2 ) 

 = ( c / )1 -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 \ (^ + ^ 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. 



