Professor Wallace's Property of the Parabola. 803 



Referring to the diagram at page 33, in the Magazine for 

 January last, (vol. x.) let p be the origin of conjugate axes, p 

 being the point of contact of the tangent A P ; and let a line be 

 drawn from A, any point in this tangent, to the focus F. The 

 angle v contained between this line and a second tangent A Q, 

 will be obtained from the equations of the lines themselves by 

 help of the known relation, 



(a — a') sin /3 

 tan v - l+aa! + , a + a i) cos/3 > 



a and a representing the coefficients of x in those equations, 

 and j3 the inclination of the axes of reference. Let the co- 

 ordinates of the point of contact Q be x n y i ; then those of the 



point A will be 0, ~ ; also, the point F is (w, — 2 m cos /3). 



SE 



Hence 



2 in 

 Equation of A Q is y = (* + *,) 



V 

 4r + 2 m cos Q 



Equation of A F is y — ^ = — x 



the tangent of the angle between them is by the preceding 

 formula 



tant; 



1-1-12 cos /3 + — cos/3-^cos/3-2cos 2 £ 



y { y, 2 m 



= \y>* m Zl = tan p. 



S^+ -p- + 2cos/3lcos/3 



Hence, since the angle § is equal to the angle F P A, it 

 follows that from whatever point in a tangent to a parabola 

 two lines be drawn? one to touch the curve, and the other to 

 the focus, the angle between them will be constant, and equal 

 to that between the fixed tangent and radius vector of its point 

 of contact. Thus F C subtends equal angles at A and B, 

 and therefore the points A, B, F, C are in the same circum- 

 ference. 



It follows moreover that, B being upon the tangent R B, 

 F B P = F R B; and since FPB= FBR .\ PFB 

 = R F B, a property which must be previously established, 

 in the geometrical method, and of which I have never seen 

 any analytical proof. 



