y3 



called the principal Jbci, and the corresponding rays the axes of the system ; determination of 

 these points and rays by means of the characteristic function, - - 46 



Each axis is intersected in its own focus by all the rays indefinitely near ; this focus belongs 

 to an osculating focal mirror, which has contact of the second order with the given mirror, at a 

 point which may be called the vertex ..... i?* 



The principal focus is also the centre of a series of spheres, which liave contact of the second 

 order with the surfaces of constant action . . . . 48. 



Examples . '. . .. ,.,* i u • • , .:.' .• • ' *9' ^^" 



,v , XI. Images Jhrmed by Mirrors. 



The image of a luminous point, formed by any given combination of mirrors, is the principal 

 focus of the last reflected system ; the image of a curve or surface, is the locus of the images of 

 points ........ 51. 



Example ; the image of a planet's disk, formed by a single mirror, is in general an ellipse ; 

 its projection on a plane perpendicular to the reflected rays is a circle, the radius of which is 

 equal to the focal length of the mirror multiplied by the semidiameter of the planet 52, 53. 



General theorem respecting the images of small objects, formed by any combination of mirrors 



54. 



There are, in general, one or more ways of placing a given mirror so as to produce an undis- 

 torted image of a planet ; the points which are to be used as vertices for this purpose, are deter- 

 mined by two relations between the partial differentials, third order, of the mirror 55. 



XII. Aberrations. 



General series for calculating the lateral aberrations by means of the characteristic function ; 

 the longitudinal aberrations do not exist for reflected systems in general ; but there are certain 

 analogous quantities, calculated in the third part of this essay . . 56, 57. 



First application ; aberrations measured on a plane which does not p^ss through either focus ; 

 the rays which make with the given ray angles not exceeding some given small angle, are dif- 

 fused over the area of an ellipse ..... 58. 



Second application ; aberrations measured on a plane passing through one focus ; the rays 

 which were before diffused over the area of an ellipse, are now diffused over a mixt-lined space, 

 bounded partly by a curve shaped like a figure of eight, and partly by an arc of a common pa- 

 rabola, which envelopes the other curve ; quadrature of this mixt-lined space, and calculation of 

 the coefficients of the result, by means of the curvatures of the caustic surface 59, 60, 61. 



Tliird application ; aberrations measured on a plane passing through a principal focus ; the 

 rays which make with the given ray a given small angle, cut the plane of aberration in an el- 

 lipse ; if the focus be inside this ellipse, the intermediate rays are diffused over the area of that 

 .-. ;,... , .... .. -i .. . . t.-...; i, ijt2)-: . ■ 



