25 



X,, tfi, z,, these latter symbols being abridged expressions for the fol- 

 lowing quantities, 



y — /i (tx -[. liy + yz) = y, , 



2 — y C«x + /3y -h yz = r,, 



and being considered as functions of a, /3, y, determined by the con- 

 ditions 



= «x, + /Sy, + yz, , 



decoy 



3?r 



3y ' SaJ' 



S/3' 

 ^•i; 



S/3jy' 



f 



(GO 



ly "^ y' 3,33y + ''' 1^ ' 



which give, after elimination, 



' V3«' "^ 3/3' "*■ V/ 3* U»' 3« "*" 3a3/3 3^ + l»¥y1^)' 



•'' VS*' S/3' + V/ 3/3 U«3/3 3« "r 3/3" 3^ + 3/33yl7y' 

 r"r — /'i!l 4- i!!!_ J. ^\}K^ ( ^'^ ^^'^ , 3«y 3»' S«t. 3/f \ 



'- U«' 3^^ + V/ 3y V3*3y 1^ + P"y "^^ + "V l^ j " 



! 



(HO 



The equation of the caustic surfaces in x, y, z, may also be de- 

 duced from the characteristic function W, by eleminating a, ^, y, 

 between the equations (/JT) and the following 



fF" = : 



or from the function V, by simply putting 



y,r = . 



(I') 



(K') 



9. The formulae of the preceding number determine by differen- 

 tiations and eliminations alone, the equation of the two caustic sur- 



VOL. XVI. 



B 



