THROUGH TRANSPARENT MEDIA, AND ON DOUBLE REFRACTION. 531 



I 



and in the result make z = 0, in order to obtain the equation of the section. This equation 

 will thus become 



{ax' + fiy-y jcL^v' + (j'y'f { a\v+(i"yy _ 



Supposing in this equation x' and y' to be referred to the axes of the section, and r, )■', to 

 be the two semi-axes, we shall have 



1 a a a 

 ^ "? "^ ¥''"? ' 



r'- " a' ■•" 6^ ■*■ c^ ' 



a/3 a'P' a"/3" 

 and = -t: + -^ + -^ . 

 or b- & 



The equation of the surface, the radius-vectors of which in a given direction are r and /, is 

 consequently the following : 



c" I Vr» a' b^ c- I 



1 l/a-'+/3' a"+(i" a"^ + /3"^ a'/3^ a' (i" a'fi'" a' (i" + a" ^' 



)•' r* V "•' 



)^'J 





"^ a'c' 



h-c' 



= 0. 



By combining with this the equations 



a' +(i' +y' = 1, 

 a'' + /3'^> + -y'^ = 1, 



iaii a'/3' a"/3' 



(?-^^°-f)=»' 



we obtain, 



, i/i-y 1-7'^ \-il\ ("/3' - «'/3)' («/3"-a"/3f («'/3" - «"/3y 



Again, from tiie equations 



a'' +a" +a"' = 1, 



(i' +(i" +i3"= = I, 



2 . '2 , "2 1 



7+7 +7 = ', 



a/3 + a'/3' + a"/3" = 0, 



|we have, 



a'(¥ + u'lr + 2«/3«'/3' = a"*/3"= = (I - a^ - a'^) (1 - /3^ - /3'') 



= ,_„=_ /3^ _„'» - /3" + a'fi* + a"li" + a'/3" + a"^'. 



Hence, o = y' - u" - fi" + (a/3' - (ia'f ; 



or, («/3' - /3a')^ = «'^ + /3'' - 7^ = I - y" -y- y'" ; 



so («/3" - a"/3)= = 7'^ and (a'/3" - a"/3'f = y'- 



