61 



This condition is independent of the magnitude of the unit of 

 distance, by which we have supposed the two planes of aberration to 

 be separated : there are therefore an infinite number of systems of 

 elHpses or hyberbolas, similar to the system {W*), and all having 

 their axes contained in the same pair of rectangular planes, which 

 pass through the principal ray : and it is natural to take these planes 

 for the planes of xz, yz, the plane of xy being still the same plane 

 of aberration as before. And thus, the intersections of these three 

 rectangular planes, may be considered as furnishing, in general, 

 three natural axes of an optical system, which are perpendicular each 

 to each, and cross in the principal focus. These natural axes possess 

 many other properties, of which we hope to treat hereafter ; but in 

 the foregoing remarks we have only aimed to shew, by some selected 

 instances, the possibility of deducing the geometrical properties of 

 optical systems of rays, from the fundamental formula (^A), 



with the assistance of the characteristic function V, and of the con- 

 nected function W : and believing that this object has been accom- 

 plished, we shall conclude the present Supplement. 



