ON SOME PROPERTIES OF THE PARA- 

 BOLA* 



THERE are many very interesting properties of the Conic 

 Sections which are not to be found in the usual works on the 

 subject, but are scattered through various memoirs in scientific 

 Journals. Those relating to the properties of polygons inscribed 

 in and circumscribed round conic sections, have been investi- 

 gated by a great many writers both in France and England. 

 Pascal was the first who engaged in these researches, and 'was 

 led by the curious properties which he discovered to call one of 

 these polygons the " hexagramme mystique." After him Mac- 

 laurin gave a proof of a theorem which is not only beautiful in 

 itself, but also very fertile in its consequences. In more recent 

 times Brianchon has demonstrated the remarkable theorems, 

 that in all hexagons either inscribed in or circumscribed round 

 a conic section, the three diagonals joining opposite angles will 

 intersect in one point. Subsequently, Davies in this country, 

 and Dandeliri in Belgium, proved in different ways the same 

 propositions along with others. The latter adopted a very 

 peculiar method, deducing these and many other properties of 

 sections of the cone by considering the cone as a particular case 

 of the " hyperboloide gauche." Generally speaking, the Geo- 

 metrical method is more easily applied than the Analytical to 

 these cases, and accordingly all the proofs given have depended 

 on geometry, with the exception of one published by Mr Lub- 

 bock in the Number of the Philosophical Magazine for August 

 1838. He has there demonstrated, by analysis, Brianchon's 

 Theorem for a circumscribing hexagon in the particular case 

 where the conic section is a parabola; but his method is tedious, 

 and not remarkable for symmetry and elegance, so that another 

 proof is still desirable. The following one is founded on the 



* Cambridge Mathematical Journal, No. V. Vol. I. p 204, February, 1839. 



