rs 



VII. Lines of reflection on a mirror. 



The curves in which the developable pencils meet the mirror, are called the lines of reflexion ; 



differential equations of these lines ; example 'i-'—T-'-'. * ' • 32,33,34 



Formulae which determine at once the foci, and the lines of reflexion ; example . 35. 



VIII. On Osculating Focal Mirrors. 



Objectof this section . . . " . . . 36. 



When parallel rays fall on a curved mirror, the directions of the two lines of reflexion are the 

 directions of osculation of the greatest and least osculating paraboloids; and the two foci of the 

 reflected ray, are the foci of those paraboloids . . . .37. 



In general the directions of the lines of reflexion are the directions of osculation of the greatest 

 and least osculating focal mirrors ; and the two foci of the ray are the foci of those two mirrors, 



38. 



The variation of the osculating focal length, between its extreme limits, follows an analogous 

 law, to the variation of the radius of an osculating sphere . " .'> . 39. 



If on the plane passing through a given reflected ray and through a given direction of oscu- 

 lation, we project the ray reflected from a consecutive point on that direction, the projection will 

 cross the given ray in the osculating focus corresponding . . . 40. 



IX. On thin and undevelopable pencils. 



Functional equation of f Am jDe««7« . . . • .41. 



When we look at a luminous point by any combination of mirrors, every perpendicular section 

 of the bounding pencil of vision is an ellipse, except two which are circular; namely, the section 

 at the eye, and the section whose distance from the eye is an harmonic mean between the dis- 

 tances of the two foci ; when the eye is beyond the foci the radius of this harmonic circular sec- 

 tion is less than the semiaxis of any of the elliptic sections . 42. 



Whatever be the shape of a thin pencil, provided it be closed, the area of a perpendicular sec- 

 tion varies as the product of the focal distances . . . 43. 



The tangent plane to an undevelopable pencil does not touch the pencil in the whole extent 

 of a ray ; it is inclined to a certain limiting plane, at an angle whose tangent is equal to a con- 

 stant coefiicient divided by the distance of the point of contact from a certain fixed point upon 

 the ray ; properties of the fixed point, the constant coefficient, and the limiting plane 44, 45. 



X. On the axes of a Rejiected System. 



The intersection of the two caustic surfaces of a reflected system, reduces itself in general to a 

 finite number of isolated points, at which the density of light is greatest ; these points may be 



