MATHEMATICAL NOTE*. 



IN Vol. I. p. 205 f, there were found for the co-ordinates of 

 the point of intersection of two tangents to a parabola, the ex- 

 pressions 



y as 0} (a -f a*}) a? = maa', 



a, a' being the tangents of the angles which the tangents to the 

 curve make with the axis of y. From these expressions it fol- 

 lows, that if y^y^ &c. x l9 # 2 , &c. be the co-ordinates of the 

 angles of any re-entering polygon of 2n sides circumscribing a 

 parabola, 



and 



Also, the continued product of the abscissae of the points of 

 intersection of any number of tangents, is equal to the continued 

 product of the abscissae of the points of contact, provided no 

 three points of intersection lie in the same straight line. 



Let a;', x" ', a;"', &c. be the abscissae of the points of contact, 

 then it is easily seen, from the equation to the parabola, that 



x' = ma!\ x" = ma."*, x'" = mv!"\ &c. 

 the continued product of which is 



x'x'x'" . . . x (n] = m W V' 2 . . . a (n)2 . 



And if x lt # 2 , a; s , &c. be the co-ordinates of the points of inter- 

 section of the tangents, we have 



Xj^ moLO.", o? 2 = wa"a"', x 3 = ma."a!" t &c. 

 the continued product of which is 



x,x,x 3 . . . x n = m* . a W" 2 . . . a (w)2 , 



which is equal to the preceding expression. It is necessary to 

 limit the intersections in such a way that no three shall lie in 

 the same line, because otherwise some one of the a's in the 

 second series would appear more than twice. 



* Cambridge Mathematical Journal, No. VII. Vol. n. p. 48, November, 1839. 

 t Page 63 of this Volume. 



6 



