282 On the Surfaces of the Second Order. 



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

 m sec $, where m is the modulus, and the angle which the 

 primary axis makes with a directive plane. 



When the umbilicar property is applied to the cone, the 

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

 licar focus, having the directive axis for its directrix, it indi- 

 cates 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 the vertex of the cone affords the only 

 instance of a focal point which is at once modular and umbi- 

 licar, 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 asymp- 

 totic limit of the two kinds of hyperboloids. For if a series 

 of hyperboloids have the same asymptotic cone, and their 

 primary axes be indefinitely diminished, they will approach 

 indefinitely to the cone; and, in the limit, the focal ellipse 

 and hyperbola of the hyperboloid of one sheet will pass into 

 the vertex and the focal lines of the cone, thus making the 

 vertex doubly modular ; while the focal ellipse of the hyperbo- 

 loid of two sheets will also be contracted into the vertex, and 

 will make that point umbilicar. 



of the distances from the given planes, we should not take the sum after multi- 

 plying the one square by any given positive number, and the other square by 

 another given positive number ; nor is there any reason why we should not take 

 other homogeneous functions of these distances. This conception would therefore 

 be found of little use in geometrical applications ; while the modular principle, on 

 the contrary, by employing a simple ratio between two right lines, both of which 

 have a natural connexion with the surface, lends itself with the greatest ease to the 

 reasonings 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, has 

 been borrowed from M. Cauchy, by whom it is employed, however, in a significa- 

 tion entirely different. Several other new terms are also now introduced, from the 

 necessity of the case. 



