in expressing the Equation of the Straight Line. 191 



cited some interest amongst the readers of the Philosophical 

 Magazine, and which treated purely by rectangular coordinates, 

 involves expressions of considerable complexity. 



Theorem. If three tangents to a parabola mutually inter- 

 sect, the circle described about the triangle formed by them will 

 always pass through the focus of the parabola*. 



The polar equation to the tangent at the point r l l of any 

 curve is 



r {cos (0-00 - sin (0-0,) ^J-| = r x . 



Edinb. Trans., vol. xii. p. 408. 



And the equation of the parabola, referred to its focus as pole 

 and diameter as origin of polar angles, is, at the point r x Q l9 



^(1+008 0!) = 2 a. 

 From (2.) we get 



dr x sin0 t 



d x ~ 1+COS0J ' 

 which, inserted in the general equation, gives at once 



r {cos (0— 0j) + cos (0-00 COS0J— sin (0— 0j) sh^} 

 = r x (1+cos0j), 



or finally, r cos (0—1 x ) cos i 0! = a (1 .) 



Similarly, r cos (0— A0 2 ) cos 1 2 = «, (2.) 



and r cos (0— i0 3 ) cos|0 3 = a, (3.) 



which represent the three tangents at t\ 1} r^ 2 , r 3 3 ; and 

 from which the proof of the theorem is deducible as follows: 



Denote by Rj 6 X the coordinates of the intersection the 

 tangents represented (2, 3), by R 2 8 2 that of (3, 1), and by 

 R 3 e 3 that of (1, 2). Then we get immediately 



* Wallace, in the Mathem. Repos., vol. ii. p. 54, Old Series, and in his 

 Conic Sections, p. 167 ; Tirnmermanns.inQuetelet's Correspondance Math, 

 et Phys., torn. ii. p. 75; Strong and Avery, Gill's Math. Misc. New York, 

 No. 6; Jones in the Gentleman's Diary, 1831; Poncelet, Traite des pro- 

 prietes projectives, section iv. Annates des Mathcmatiques, tom. viii.; Phil. 

 Mag., S. 3, vol. ix. p. 100; x. pp. 32, 35; xi. p. 302; and Young's Conic 

 Sections, p. 189. 



I would not be understood to contest the simplicity of the geometrical 

 methods of proving this theorem ; but merely take this theorem as an il- 

 lustration of the occasional advantage of the polar over the rectangular 

 equation of the tangent to a curve. 



