30 



W + W + 4 " ^' + i^ SW 4- &c. : 



in which 



&c. 



The equations 



w' = w, iw = iw, (TO 



will be satisfied independently of the ratios of the variations S«, S0, Sy, 

 if we take the point X, Y, Z, any where upon the given ray, and 

 suppose. 



There remains therefore one arbitrary constant of the focal system to 

 be determined, and this is to be done by equating the next terms of 

 the developments, that is by putting 



^^w' = yw, (U') 



and assigning some limiting ratios to the variations So., 3^, iy, con- 

 sistent with the differential equation 



■ i 



which results from a^ + /3^ + y^ = 1 . And, from the nature of the 

 functions W, ff', the equation ([/') may be put under the following 

 form : 



