99 



SfV _ S'v rv i'v 



SPF rv rv r-v 



= '=T:%7 + yT^ + ^ 



between which the three quantities a, j8, y, are to be eUminated ; TV 

 being an arbitrary but homogeneous function of these three quanti- 

 ties, of the dimension zero ; and the partial differential coefficients in 

 which the sign S occurs, being formed by differentiating the homoge- 

 neous functions fV, v, as if a, /3, y, were three independent variables. 

 In applying these general results to ordinary systems of rays, we are 

 to put 



w = f.(«^ + ^^ + y»)*; 

 Sv jM-'a St) __ j/,"/} Sr [x.^Y 



^v u^ „ „ .„, pv 



,2 



£1 _ tl r«' — L<,2 «M -fiL - ^ („2 _ <,= an i-^ - iil (^ _ u,^ v«i ■ 



f^ ~ ?~ ' 3/33y v^ ' "hyla. ~ ~ ' 



or, (making after the differentiations a* -f- /S* + y* = 1,) 



Ju Iv Iv 



|^=^(l_«"-), 1^=^(1-^^). |^=f.(l_y.). 



S'^u 3=u 3'u 



= -."-'"^> s^is:^ = — f'^y ' ^;a:=— f*y«i 



3«J.3 - ■'' ' 3.5Sy - '^''^ ' 3yS, 



# 



