50 Mr. G. Salmon on Theorems of Pascal aud Brianchon. 



demonstration of Pascal's I would refer to Gergonne's An 

 nales, vol. xvii. p. 222. 



I venture, however, to offer in addition the following proof 

 of Brianchon's theorem, because it leads at once to the corre- 

 sponding property in surfaces of the second degree. 



Let S = be the equation of a given conic, L = that of 

 a right line ; then it is easy to see that S — L 2 = is the 

 equation of a conic touching the given in the two points where 

 it is met by the line whose equation is L = 0. 



Let S — L 2 = be the equation of another cone having 

 also double contact with the given. Subtracting these equa- 

 tions, we have for the intersection of the last two conies 

 L 2 — L 2 = 0. The equations therefore of two chords of in- 

 tersection are L — L y = 0, L 4 L / = 0. These two chords 

 must pass through the intersectionsofthetwo chords of contact, 

 since their equations are satisfied by the combined equations 

 L = 0, L y = 0. 



It would lead me into too much detail to prove that the 

 form of the equations shows that the four lines form an har- 

 monic pencil. 



Now let a third conic have also double contact with the 

 given. Its equation will be of the form S — L /y 2 = 0. The 

 equations of its chords of intersection with the other two, 

 L - L» = 0, L + L„ = 0, L, - L„ = 0, L, + L„ = 0. 



Evidently the three equations L — L, = 0, L ; — h,, = 0, 

 L — L 7/ = are satisfied for the same point ; also L — L t = 0, 

 L y + L y/ = 0, L + L w as 0, and so of the rest. Hence "if 

 three conies have each double contact with a fourth, their six 

 chords of intersection with each other pass three by three 

 through the same points." 



Now let each of the three touching conies degenerate into 

 a pair of right lines, and we have Brianchon's theorem. 



Now everything we have said applies almost word for word 

 to surfaces. 



The equation of a given surface of the second degree being 

 S as 0, that of a plane Jj = 0, another surface touching the 

 given along this plane will have its equation of the form 

 S — L 2 = 0. A second surface also enveloped by the given 

 has for its equation S — L/ 2 = 0. Precisely as before these 

 two surfaces intersect each other along the planes whose equa- 

 tions are L — L, =s 0, L -f- L, = 0. Hence " if two surfaces 

 of the second degree are enveloped by a third they will inter- 

 sect each other in two plane curves, and the planes of inter- 

 section pass through the intersection of the planes of contact." 

 Let a third surface be also enveloped by the surface S; its 

 equation is of the form S — L y/ 2 = 0, and its planes of inter- 



