MATHEMATICAL NOTES*. 



1. IN the Examination Papers for 1834, the following 

 problem is given: "If the chord of a conic section, whose 

 eccentricity is e, subtend at its focus a constant angle 2a, prove 

 that it always touches a conic section having the same focus 

 whose eccentricity is e cos a." A solution of this problem by 

 a peculiar analysis will be found in a preceding article ; but the 

 following method may be found not uninteresting. 



Let r l5 r z be radii vectores to the ends of the chord, < a, 

 < 4- a, the corresponding angles vectores, p the perpendicular 

 from the focus on the chord ; 



.'. p x chord = r^ sin 2a, 

 . lV^ + r, 2 - 2r,r > cos2a) 



' p rfa sin 2a 



// 1 1 2 \ 



= cosec 2a / f -jH 2 cos 2aj , 



V * 1 2 12 ' 



1 _ 1 H- e cos (^ a) JL <fr 



For cos 2a put 12 sin 2 a ; then, by a few obvious steps, 



- = , fJ(l + 2e cos a cos <b + e 2 cos 2 a). 



^? t cos a 



Put a = ; then the chord becomes the tangent, and 

 11 



Po I 



But the general form coincides with this, if we put 



I cos a = X and e cos a = e ; 

 for then 



* Cambridge Mathematical Journal, No. IX. Vol. in. p. 94, February, 1844. 



