21 



responding changes in the coefficients of the function fV, by means of 

 the relations which we have already pointed out, between these two 

 characteristic functions ; observing, that while the value of V itself 

 is not altered in the act of reflexion or refraction, but only its form 

 and its difl^erentials, the value of IV receives a sudden increment, 

 which has for expression, 



aW'=:xa i— +yAi— 4-za c— 



(hi , iu 3m \ 



hj ^ '■^zJ- <Y) 



7. By the help of the foregoing formulae, we can compute the 

 partial differential coefficients of any given order, of the characteris- 

 tic functions V and JV, for any homogeneous system of straight rays, 

 produced by any finite number of successive reflexions and refractions 

 ordinary or extraordinary, when we know the nature of the light and 

 of the mediums, and know also the coordinates of the luminous origin 

 and the equations of the reflecting or refracting surfaces. To shew 

 this more fully, let us observe, that in a system of straight raj^s di- 

 verging from a luminous point, and not yet reflected or refracted, we 

 may put 



J being the distance from the luminous origin X, Y, Z, to any other 

 point X, J/, z; and that we have the equations, 



• ftliu 

 from which we can deduce the partial diftbrentials of the functions 



V and TV, in this first state of the system ; those of the second order, 



ibrexample, being given by the following expressions: 



