116 



gether with the corresponding vertex upon the last mirror of the given combination ; the rayg 

 which come to this given vertex, from the several points of the object, pass after reflection 

 tlirough the corresponding points of the image. 



[_55,'2 It follows from this theorem, that in order to form, by a single mirror, an undistorted 

 image of any small plane object, whose plane is perpendicular to the incident rays, it is neces- 

 sary and sufficient that the plane of the image be perpendicular to the reflected rays. This 

 condition furnishes two relations between the partial differential coefficients, third order, of the 

 mirror, which will in general determine the manner in which the object and mirror are to be 

 placed with respect to one another, in order to produce an undistorted image. Thus, if it were 

 required to find, how we ought to turn a given mirror, in order to produce a circular image of a 

 planet ; we should have the following condition, 



d^ = x'dx + li'di/ + yrfz, (H") 



which expresses that the reflected rays are perpendicular to the plane of the image ; «', ,8', y', 

 being the cosines of the angles which the incident ray makes with the axes of coordinates; and ^ 

 being the focal length of the mirror, which by [49,] is equal to half the geometric mean be- 

 tween the radii of curvature ; so that it is a given function of the partial differentials, first and 

 second orders, of the mirror, 



the cosines («, ,3', y') may also be considered as given functions of (p, g, r, s, t), because, by 

 [49.] the incident ray at the vertex is contained in the plane of the greatest osculating circle to 

 the mirror, and the square of the cosine of angle of incidence is equal to the ratio of the radii of 

 curvature. The two equations therefore, into which (H") resolves itself, by putting separately 

 dy = 0, dx =: 0, will furnish two relations between the partial differentials of the mirror, up to 

 the third order ; these are the two relations which express the condition for the image of the 

 planet being circular : they are identically satisfied in the case of a spheric mirror, for then the 

 first member of (H") vanishes, on account of the focal length being constant, and the second 

 member on account of the incident ray coinciding with the normal; and accordingly, whatever 

 point of a spheric mirror we choose for vertex, it will form a circular image of a planet ; but 

 when the mirror is not spheric, these two relations will in general determine a finite number of 

 points upon it, proper to be used as vertices, in order to form an undistorted image. And when 

 we shall have found these points, which I shall call the Vertices of Circular Image, it will then 

 remain to direct towards the planet, one of the two lines which at any such vertex are contained 

 in the plane of the greatest osculating circle to the miiTor, and which make with the normal, 

 at either side, angles, the square of whose cosine is equal to the ratio of the radii of curvature. 



