138 



that pass through any given small area on the plane of aberration, are diffused upon another 

 perpendicular plane, which crosses the given reflected ray at the point where that ray meets the 

 mirror, we are to employ these other formulae (see [58.]) 



a, b, being the coordinates of the point in which a near ray crosses this latter plane, and ^,, ^j, 

 the distances of that point from the two caustic surfaces, that is, the two focal lengths of the 

 mirror. In this manner we find, that to any given point (x', i/) on the plane of aberration, cor- 

 respond two other points on the other perpendicular plane, determined by the equations 



«=--^, 6 = -^. (Bx'zfV[^Ci^2/'+(B^-^C)x'']); (Q'") 



understanding however thai these two points become imaginary, when the quantity under the 

 radical sign is negative, that is, when the point (x', y) is at the wrong side of the enveloping 

 parabola (jR"), [60.]; which parabola, within the small extent in which it is taken, may be 

 considered as confounded with the normal section of the caustic surface made by the plane of 

 aberration. Now, if we consider any infinitely little rectangle upon this latter plane, having for 

 the coordinates of its four comers 



1st. x', y', 2d. x' + dx!, y', 3d, x', y + dy', 4th. x' + dx', jj + ^y', 



the rays which pass inside this little rectangle are diffused over two little parallelograms on the 

 other perpendicular plane ; the four corners of the one having for coordinates, 



1st. a, b, 2d. a + da, b + —p • duf, 3d. a, b ■\- -j- . dy', 



. t . , , i db , , db , , 



4th. a + rf(7, i +--j^ . rfar' + ^. dy', 



and the four corners of the other having for coordinates, 



db' 

 1st. a, b, 2d. a + da, i'-\--jJ- ^^'> 



3d, a, b' + ^,- «(y'. *th. a + da, bf + ^ • '^•^' + "T"/ • '^i^' 



b, V, being the two values of (5) given by the formula (Q'")- The areas of these two parallelo- 

 grams are each equal to (da.—r-^ , dy" ^ that is to 



gi-gB- dx'Jy' 



V [2C2y + (B^ — AC)x'^ ' 

 and the area of the little rectangle on the plane of aberration is dx'.dy' ; if then we denote by 



