coefficients of ^,r, di/, Sz, should also vanish, and furnishes thereby 

 the following general differential equations of a ray, 



of which any two include the third. And rejecting the evanescent 

 quantities in the expression for dfvds, we find the formula {A), 

 which it was required to demonstrate. 



3. The fundamental formula thus obtained, resolves itself into 

 the three followmg equations : 



(C) 



representing, for abridgment, the definite integral fvds by F, and 

 considering this integral as a fiyiction of x, y, z, of which the form 

 depends upon the nature of the system, the medium, and the light, 

 and of which the partial differential coefficients of the first order are 



denoted by 



ST SF SF ,, 

 ix ' ,iy ' ^z ' 



When the form of V is given, we can obtain these coefficients by 

 differentiation ; and if we know also the form of v, which depends 

 only on the nature of the medium and of the light, we can by the 

 equations (C) determine a, (3, y, as functions of x, y, z; that is, we 

 can find the direction of the ray or rays passing through any pro- 

 posed point of the system. The geometrical properties of one system 

 •as distinguished from another, for any given medium and any given 

 kind of light, may therefore be deduced by analytic reasonings from 

 the form of the function V ; on which account we shall call this 



VOL. XVI. c 



