50 



Natural Axes of a System. 



20. The equations, (S^*) or (T'*), express the connexion between 

 the coordinates a?, y, of the intersection of a near ray with the plane 

 of aberration passing through the principal focus, and the coordinates 

 a, h, of the intersection of the same near ray with another plane of 

 aberration, parallel to the former, and at a distance from it equal to 

 unity : they serve therefore to resolve the questions that have reference 

 to this connexion. The most interesting questions of this kind, are 

 those which relate to the comparative condensation of the near rays, 

 in crossing the two planes of aberration. Let us therefore consider 

 an infinitely small rectangle ^aM on the plane of a, 6, having for the 

 coordinates of its four corners, 



the rays which pass inside this little rectangle, will, at the plane of 

 cXJ/, be diffused over a little parallelogram, of which the coordinates of 

 the corners are 



• I".,y;II-.+ ^±3.,, + |3«;iii.. + |3,., + |s6; 

 the partial differential coefficients 



Ix 3j; 3y 3y 

 ia"' ^' 51' ib*^ ' 



being obtained by differentiating the equations [S*), or (T*). The 

 area of the parallelogram on the plane of xy is 



I 2 



.■i\ Jii.iJ 



