74 



curve ; otherwise they are diffused over a mixt-lined space, bounded partly by an arc of the el- 

 lipse, and partly by two limiting lines, namely the tangents drawn from the focus . 62. 



The distinction between these two cases depends on the nature of the roots of a certain qua- 

 tain quadratic equatiou ; when the focus is inside the ellipse, the caustic surfaces do not intersect 

 the plane of aberration ; but when the focus is outside, then the caustic surfaces intersect that 

 plane in the limiting lines before mentioned. . . . 03. 



This distinction depends also on the roots of a certain cubic equation ; when the focus is in- 

 side there are three directions vli focal inflexion on the mirror, and of spheric inflexion on the 

 surfaces of constant action, but when the focus is outside, there is but one such direction ; 

 the aberrations of the second order vanish, when there is contact of the third order between the 

 mirror and the focal surface, or between the surfaces of constant action and their osculating spheres, 



64, 65. 



XIII. Density. 



Method by which Malus computed the density for points not upon the caustic surfaces ; 

 other method founded on the principles of this essay ; along a given ray, the density varies in- 

 versely as the product of the focal distances ; near a caustic surface it varies inversely as the 

 square root of the perpendicular distance from that surface . . 66, 67, 68. 



Law of the density at the caustic surfaces ; this density is greatest at the principal foci, and at 

 a bright edge, the locus of the points upon the caustic curves, at which their radius of curvature 

 vanishes ...... 69, 70. 



Density near a principal focus ; ellipses or hyperbolas, upon the plane of aberration, at which 

 this density is constant; the density at the principal focus itself, is expressed by an elliptic inte. 

 gral, the value of which depends on the excentricity of the ellipses or hyperbolas, at which the 

 density is constant . . . 7 1 , 72, 73, 74, 75, 76. 



PART SECOND. ON SYSTEMS OF REFK ACTED RAYS. 

 XIV. Analytic expressions of the lata of ordinary refraction. 

 Fundamental formula of dioptrics ; principle of least action ; cartesian surfaces, 77, 78, 79. 

 XV. On Jbcal refractors, and on the surfaces oj" constant action. 



Differential equation of focal refractors ; this equation is integrable, when the incident rays 

 are peipendicular to a surface ; form of the integral ; the focal refractor is the enveloppe of a 

 certain series of cartesian surfaces .... 80. 



When homogeneous rays have been any number of times reflected and refracted, they are cut 

 perpendicularly by the surfaces of constant action . . .81. 



