medium and of the light is known. Reciprocally, if the connexion 

 between W, a, /3, y, be given, that which exists between F, x, y, z, 

 can be found. For if we suppose that for the sake of symmetry, W 

 has been put under the form of a homogeneous function of the 

 dimension i, by the help of the relation a* + /S* + 7' = 1, and then 

 differentiated as if a, (3, y, were three independent variables, we shall 

 have, by (£), and by the nature of homogeneous functions, 



__^ ,^<, +^ __ + y __+ , __. 



in which we shall for simplicity suppose the dimension ^ = ; and 

 eliminating a, /3, y, by means of these equations, from that marked 

 CD), we shall deduce the relation between F, x, y, z, from the rela- 

 tion between W, a, /3, y. We may therefore consider W as itself a 

 characteristic function, which distinguishes any one homogeneous 

 system of straight rays not parallel, from any other such system, com- 

 posed of light of the same kind, and contained in the same medium. 

 It is evident that on some occasions it must be advantageous to attend 

 to the function W instead of F, because F changes in passing from 

 one point to another of the same ray, whereas W is constant, when 

 the ray and the system are given. On the other hand, in any sud- 

 den change of the system by reflection or refraction, the function W 

 receives a sudden alteration, while the change of F is gradual ; it is 

 therefore convenient to employ F instead of W, in investigating the 

 effects of such changes. Accordingly, in the remainder of this 

 memoir, we shall consider both these functions, and examine the 

 relation between them : and shall begin by investigating the con- 

 nexions between their partial differential coefficients. 



c 2 



