53 



XiyVii z,, and ^x\ iy', ix!, having the same meanings as before ; and 

 the condition of minimum gives 



•■ — a«» + ^^ +v 

 ■which may also be thus written 



= Wx' + S/3V + SySz' = 5«^^ + ^^h + ^y^^ = ^"^'^ 



or, by the foregoing number, 



(li^+^+ Vr^=^ S^^i^"*"^ S^^S^+^ Sy ^ 3-'- ^^ 



Another transformation of this condition, vi'hich shall involve the 

 function W instead of V, may be obtained in the following manner. 

 Let W, be the form which the characteristic function W would have, 

 for a system of rays of the same light and in the same medium, but 

 all converging towards or diverging from the one point x,, y,, z,; so 

 that, by the theory already given. 



i3y "*■ y'" i$iy 



then, by diflferentiating the equations (G'), and attending to the for- 

 mula (F"0, we find 



= « (Jx' — r5«) + ,3 (V — rJ/S) + y (Jz' — rSy) ; 



