118 



(O") 



and these forms for tbe coefficients of the development {H"), agree 

 perfectly with the development (C") or (E"), when we establish the 

 following relations between the constants : 



Do = — WW ;v,=z2A;v,=— I6iv, ; v^ = 8 (IQw, — 8w, — A) : (P") 



we see, therefore, that the present method of integration confirms the 

 former results. 



Connexion of the Longitudinal Aberration in a System of Revolution, 

 with the Development of the Characteristic Function V. 



9. To give now an example of the optical use of the development 

 which has been thus obtained, let us consider its connexion with the 

 aberrations of the near rays, from the principal oc central focus. We 

 have already remarked that the constant A denotes the distance of 

 the origin of coordinates, upon the central ray, beyond this principal 

 focus, in such a manner that the focal ordinate is = — A. For the 

 ordinate Z, of intersection of any near ray with the central ray, we 

 have by the fourth of the equations (C), of the present Supplement, 



z^-^l—lf, (Q") 



if we form the coefficient ^ by putting W under the form of a homo- 

 geneous function of a, /S, y, of the dimension zero, with the help of 

 the relation a* + /3* + y* = 1, and by then differentiating this 

 function, and if «, /3, were constant, and y the only variable. 



