469 



point from two directive planes drawn througli the direc- 

 trix corresponding to that focus ; and it is easy to see that 

 this ratio, the square root of which we shall denote by ju, is 

 equal to L — M, or, neglecting signs, to the sum of the nu- 

 merical values of l and m. Of course, if the distances from 

 the directive planes, instead of being perpendicular, be 

 measured parallel to any fixed right line, the ratio will still 

 be constant, though different. For example, if the fixed 

 right line for each plane be that which joins the corres- 

 ponding umbilic with either focus of the section xy, the ratio 

 of the square to the rectangle will be the square of the 

 number »z sec ^, where m is the modulus, and ^ the angle 

 which the primary axis makes with a directive plane. 



When the umbiHcar property is applied to the cone, the 

 vertex of which is, as we have seen, to be regarded as an 

 umbilicar focus, having the directive axis for its directrix, it 

 indicates that the product of the sines of the angles which 

 any side of the cone makes with its two directive planes is a 

 constant quantity. 



It is remarkable that tlie vertex of the cone affords the 

 only instance of a focal point which is at once modular and 

 umbilicar, as well as the only instance of a focal point which 

 is doubly modular. This union of properties it may be con- 

 ceived to owe to the circumstance that the cone is the 

 asymptotic limit of the two kinds of hyperboloids. For if a 



with the surface, lends itself with the greatest ease to the i-easonings of geometry. 

 Indeed the whole difficulty, in extending the property of the directrix to surfaces 

 of the second order, consisted in the discovery of such a ratio inherent in all 

 of them ; a ratio having nothing arbitrary in its nature, and for which no other 

 of equal simplicity can be substituted. 



It may be proper to mention that the term modulus, which I have used for 

 the first time in the present paper, with reference to surfaces of the second 

 order, lias been borrowed from M. Cauchy, by whom it is employed, however, 

 in a signification entirely different. Several other new terms are also now in- 

 troduced, from the necessitv of the case. 



