LXXVI. On a Neva Class of Properties of Lines and Surfaces 

 of the Second Order. By the Rev. J. Booth, LL.D.f 

 M.R.I. A., Vice-Prijicipal of and Pre fessor of Mathematics 

 in the Liverpool Collegiate Institutiofi^P'*-^ ^- ,.ao];jog ainoj 



IT lias sometimes been made an object of inquiry with geo- 

 meters, whetiier foci and directrices exist as well for the 

 minor as the major axes of a conic section; and the conclu- 

 sion generally acquiesced in by mathematicians seems to be, 

 that for the minor axes these points and lines are imaginary ; 

 this it would appear proceeds from assuming as a definition of 

 a conic section, some property which, instead of being funda- 

 mental, is merely a particular case of a more general theo- 

 rem ; thus the definition of Boscovich, which is usually 

 adopted as the fundamental definition in elementary analyti- 

 cal treatises on this subject at the present day, that a conic 

 section is the locus of a point whose distances from a fixed 

 point, and from a given line, are in a constant ratio, is merely 

 a simplified case of the following more general theorem : — 

 That if in any siirface of the second order isoith three unequal 

 axes, two planes {called directrix planes) are drawn through a 

 certain line, parallel to the circular sections of the surface, and 

 if a certain point he assumed [which may he termed the focus 

 of the surface), the square of the distance of any point on the 

 surface fom this focus, hears a constant ratio to the product of 

 the perpendiculars from this point on the two directrix planes. 



When the surface is one of revolution round the transverse 

 axe, the two groups of circular sections becoming coincident 

 in direction, the two directrix planes coalesce, and the perpen- 

 diculars are coincident and equal; hence the above quadratic 

 relation may be depressed to the common linear condition be- 

 tween the focus, directrix plane and any point on a surface of 

 revolution round the transverse axe, or on ,a section of this 

 surface passing through it. '"' ' . .,•« * . ,,-i , ■ 



When the surface is an oblate sphertilrfj'^tftfe^Wfe^ttM'jimiifes 

 are in this case also identical ; the point termed the focus co- 

 incides with the centre; and if any central plane be drawn, it 

 will cut the surface in a conic section and the directrix plane 

 in a right line, which is lernied in the following paper the 

 minor directrix of the conic section, 

 wjn Another definition of a focus has been assumed, that the 

 ftjcus is a point in the plane of the curve whose distance from 

 any point on the curve is a linear function of the correspond- 

 ing co-ordinates; and in a very elaborate and masterly paper, 

 published in the number of the Philosophical Magazine for 



* Communicated by the Author. 



