Dr. Booth ofi a New Clas^ of Properties of Curves. 539 



January 1843, Mr. Davies ttieftts the question *of directrices 

 , and foci in a manner perfectly general, and arrives at the con- 

 clusion, that there are but two foci and two directrices in a 

 conic section, a result arising from the arbitrary nature of the 

 definition which he had assumed as the basis of his investiga- 

 "'tion; a definition to which it would appear there lie two dj- 

 jectibns ; first, that it is merely an analytical condition having 

 "tio geometrical representation, at least such a one as would 

 show that these points should occupy the important position 

 they do in the theory of conic sections, a fact which becomes 

 ^'^t once manifest, when these points are defined as the points 

 ~'df contact of a plane touching two spheres inscribed in a right 

 cone, and which cuts it in a conic section ; and again, because 

 'the distance of the focus to any point of the common direc- 

 trix, or of the centre to any point of the minor directrix, is a 

 linear function of the co-ordinates of the point of contact of a 

 tangent drawn through this point of the directrix to the curve, 

 so that other points may be found which are not on the curve 

 whose distances to the focus are linear functions of the co-or- 

 ^'dinates of certain corresponding points upon the curve. 

 ^ It is true, that so long as a conic section is defined as a 

 ^ plane section of a right cone, the minor directrices cannot be 

 ' exhibited by any construction at all comparable in elegance 

 with the geometrical method of defining the common foci and 

 '"directrices above alluded to; but if we define a conic section 

 as a central plane section of any surface of the second order, 

 the difficulty at once vanishes, and we can exhibit the minor 

 directrices with as much ease as we may the ordinary foci and 

 directrices: to give some of the leading properties of the 

 former is the object of the following paper. nii -e-iiilupib 



• - " Let « and 6 be the semiaxes of a central conic sectiohj <? the 



eccentricity = ( - — 2~") ' ^^'^j g"'ded by the analogy of the 

 ordinary directrices and foci, let us draw two right lines per- 

 pendicular to the minor axe, at the distance — froni the cen- 



e 



tre, these lines may be termed the minor directrices^, and on 

 the same axe let two points be assumed at the distance b e 

 from the centre, these points may be called the minor foci. 



In the hyperbola, as the minor axe is imaginary, the new 

 directrices must be drawn in a somewhat different manner, 

 Let 2 w be the angle between the asymptots of the hyperbola, 

 and on the transverse axe let two points be assumed at the 

 distance a sin co from the centre ; through these points let per- 

 pendiculars to the transverse axe be drawn, these lines are the 



2 N 2 



