28 





the equations [B') become, 



3 w i^v rv , „. s'v 



i» d» ' Sad/i ' detdy 





in 

 iw 



Oy 





(P') 



,.yv 



^.. Ja3y ^ » 3/33y *-^ 



3y " 



Now, if we conceive another system of rays, composed of the 

 same kind of Hght, and contained in the same medium, but all con- 

 verging to or diverging from the one point, a/', y", z", and represent 

 by W', the characteristic function, which, in this new system, cor- 

 responds to W in the old, we shall have 





+ y 3«3/3 + "^ 53; 



oy 



,, J'f 



= '- 1^, + y" ^ + '" 



yv 



3^» 



^'iy' 



enoy 



= «■' ^r-r + y" i7<- + =' 



3/3 3y 



3yl 





,3iP 



3/3 



3«3^ 



3-^=x"J^_ + y"3"j-f + 2' 



3^ 



3,33y ' 



3y 3«Jy '^ ■' *''!*«' ^^ ■*-■* 



3/33y 



Jy» 



(QO 



the equations which determine the developable pencils, and the caus- 

 tic curves and surfaces, may therefore be thus written : 



'^"^ ' '■' '-'' '\W , Sr' .3^ . 3Jr' . 3fF __ . i^P 



"d 



