115 



equations (C"), (D"), combined with the identical relation eCcU' + ^d^ + y'dy' = 0, and with 

 the following formula, 



<?«'«-f-6?(3"'-t-(fy«=<rS (E*) 



in which <r is the semidiameter of the planet, contain the solution of the question ; for they de- 

 termine the image of any given point upon the disk ; and if we eliminate dal, d0, c?y', between 

 them, we shall find the two relations between da, db, dc, which belong to the locus of those 

 images, that is, to the image of the disk itself. 



£53.] To simplify this elimination, let us take the central reflected ray for the axis of (2), 



— ct' 



that is, letusput« = 0, ^ = 0, y=l. We shall then have by (A") p= , , 9 = 



1 + V 



— , and the formulae (C") will become 



jdy 



1+y' 



,d.' + da = ^, ,d,' + db = -^ 



which give, by the identical relations 



»" + /3«' + y« = 1 , «.'.d*' + /3'.rf/3' +y'.rfy' = 0, 

 ^dyf=a:da + ^'db 



,d.=-da^ -'-^'''';;^/''' I ,F") 



,d,'=-db+''-^''''"+''"'^ 



:0 ) 



1+y' 



Eliminating da,', rf/3', dy', by these formula;, from the equations (D") and (E"), we find, for 

 the equations of the image, 



1st. (A — x').da+{B — fi').db'tCdc: 



2d. da^ -^ db^ := f. <r^ 



the image is therefore, in general, an ellipse, the plane of which depends on the quantities 

 A, B, C, which enter into the 1st of its two equations, and therefore on the partial differentials 

 of the mirror, as high as the third order ; but the 2d of its two equations (C"), is independent 

 of those partial differentials, and contains this remarkable theorem, that the projection of the 

 image of the disk, on a plane perpendicular to the reflected rays, is a circle, whose radius is 

 equal to the focal distance (5), multiplied by (o-) the sine of the semidiameter of the planet. 



[54.] The theorem that has been just demonstrated, respecting the projection of a planet's image, 

 is only a particular case of the following theorem, respecting reflected images in general, which 

 easily follows from the principles of the preceding section, respecting the axes of a reflected 

 pystem. This theorem is, that if we want to find the image of any small object, formed by any 

 given combination of mirrors, and have found the image of any given point upon the object, to- 



