iir 



XII. Aberrations. 



[56,] After the preceding investigations respecting the two foci of a reflected ray, or points of 

 intersections with rays infinitely near ; and respecting the axes of a reflected systeno, each of which 

 is intersected, in one and the same point, by all the rays that, are infinitely near it ; we come now 

 to consider the Aberrations of rays at a small but finite distance : quantities which have long 

 been calculated for certain simple cases, but which, have not, I believe, been hitherto investi- 

 gated for reflected systems in general. 



[57.] When rays fall on a mirror of revolution, from a luminous point in its axis, the re- 

 flected rays all intersect that axis, and the distances of those intersections from the focus, are 

 called the longitudinal aberrations. But in general, the rays of a reflected system do not all in- 

 tersect any one ray of that system ; and therefore the longitudinal aberrations do not in general 

 exist, in the same manner as they do for these particular cases, which have been hitherto con- 

 sidered. However I shall shew, in a subsequent part of this essay, that there are certain other 

 quantities which in a manner take their place, and follow analogous laws : but at present I shall 

 confine myself to the lateral aberrations measured on a plane perpendicular to a given ray, of 

 which the theory is simpler, as well as more important. 



Let therefore, [x', y', z') represent the coordinates of the point in which the plane of aber- 

 ration is crossed by any particular ray ; these coordinates may be considered as functions of any 

 two quantities which determine the position of that ray; for example, of the cosines of the an- 

 gles which the ray makes with the axes of {x) and (y). They may therefore be developed in 

 series of the form 



/ ry dZ , dZ , , \ d'^Z „ ^ d'-Z d^Z „ 7 



X, Y, Z, being their values for the given ray, that is the coordinates of jthe point from which 

 aberration is measured, aud {»„ /I,') being the small but finite increments which the cosines 

 («, ^) receive, in passing from the given ray to the near ray. These equations (K") contain the 

 whole theory of lateral aberration ; but in order to apply them, we must shew how to calculate 

 the partial differential coefficients of {X, Y, Z), considered as functions of («, /3). For this 

 purpose I observe, that «, /3, being themselves the partial differential coefficients of the charac- 

 teristic of the system, (Section V.), may be considered as functions of the coordinates a, b, of 

 the projection of the point in which the ray crosses any given perpendicular surface v 



