•3 . 



their distances from any infinitely near point upon the mirror. If therefore we construct An fell' 

 lipsoid of revolution, having its two foci at the two assumed points, and its axis equal to the bent 

 path traversed by the light in going from the one point to the other, this ellipsoid will touch the 

 mirror at the point of incidence. Hence it may be inferred, that every normal to an ellipsoid of 

 revolution bisects the angle between the lines drawn to the two foci ; and therefore that rays is- 

 suing from one focus of an ellipsoid mirror, would be reflected accurately to the other. 



[5.] These theorems about the ellipsoid have long been known ; to deduce the known theo- 

 rems corresponding, about the hyperboloid and plane, I observe that from the manner in which 

 the formula (D) has been obtained, we must change the signs of the distances, j, j', if the as^ 

 sumed points X, Y, Z, X', Y', Z', to which they are measured, be not upon the rays themselves, 

 but on the rays produced. If therefore we assume one point X, Y, Z, upon the incident ray, 

 and the other point X', Y', Z', on the reflected ray produced behind the mirror, the equation 

 (D) expresses that the difference of the distances of these two points from the point of incidence, 

 is the same as the difference of their distances from any infinitely near point upon the mirror ; so 

 that if we construct a hyperboloid, having its axis equal to this difference, and having its foci at 

 the two assumed points, this hyperboloid will touch the mirror. The normal to a hyperboloid 

 bisects therefore the angle between the line drawn to one focus, and thie produced part of the 

 line drawn to the other focus ; from which it follows, that rays issuing from one focus of a hyper- 

 boloid mirror, would after reflection diverge from the other foctis. A plane is a hyperboloid 

 whose axis is nothing, and a paraboloid is an ellipsoid whose axis is infinite ; if, therefore, rays 

 issued from the focus of a paraboloid mirror, they would be reflected parallel to its axis ; and if 

 rays issuing from a luminous point any where situated fall upon a plane mirror, they diverge after 

 reflection from a point situated at an equal distance behind the mirror. These are the only mir- 

 rors giving accurate convergence or divergence, which have hitherto been considered by 

 mathematicians ; in the next section I shall treat the subject in a more general manner, and 

 examine what must be the nature of a mirror, in order that it may reflect to a given point the 

 the rays of a given system. 



II. Theory of focal mirrors, 



[6.} The question, to find a mirror which shall reflect to a given focus the rays of a given syg. 

 tem, is analytically expressed by the following differential equation, 



(«-f.«') dx + (/SH-^') dy + (y-f-y-) dz ==0, (E) 



X, y, z, being the coordinates of the mirror, and «, /3, y, «', ^', y, representing for abridgment 

 the cosines of the angles which the incident and reflected rays make with the axes of coordinates. 

 In this equation, which follows immediately from (C), or from (B),; «, /3, y, are to be considered 

 as given functions ofx,y, z, depending on the nature of the incident system, and «', fi', y, as 

 Other given functions o(x, y, z, depending on the position of the focus; and when these func. 



