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unity, there results a corresponding connexion between the partial 

 differential coefficients to which they are proportional. This con- 

 nexion is expressed by an equation which it is interesting to study 

 and to integrate, because it contains a general property of ordinary 

 systems of rays, and because its integral is a general form for the 

 characteristic function of such a system. The integral which I have 

 given in the present memoir, is deduced from equations assigned in 

 my former Supplement ; an elimination which had been before 

 supposed, being now effected, by the theorems which Laplace has 

 established in the second Book of the Micanique Celeste, for the 

 development of functions into series. The development thus ob- 

 tained, proceeds according to the ascending powers of the perpendi- 

 cular distances of a variable point from the tangent planes of the 

 two rectangular developable pencils which pass through an assumed 

 ray of the system, and according to the descending powers of the 

 distances of the projection of the variable point upon the assumed 

 ray, from the points in which that ray touches the two caustic sur- 

 faces. In the case of rays contained in one plane, or symmetric 

 about one axis, the partial differential equation takes simpler forms, 

 of which I have assigned the integrals, and have given an example of 

 their optical use, by briefly shewing their connexion with the longi- 

 tudinal aberrations of curvature. I hope, in a future memoir, to 

 point out other methods of integrating the general equation for the 



