IN THE NEIGHBOURHOOD OF A CAUSTIC. 381 



of these lines will be equal (ultimately) to the shortening XZ' of the 

 other, and their sum (ultimately) will not be altered; or that, putting 



—^ — for the differential coefficient of V with regard to x, considering 



y also as a function of x, (which is otherwise written '^— + — . — \, 



V dx dy dxl 



-T-— = 0. This is the condition which holds at the point of reflection. 



3. Now if p, q, be the co-ordinates of a focus, since in that case it 

 is a point in the paths of rays reflected from every point of the sur- 

 face, —. — = at every point, and therefore V = constant, and '^4 — -, 



ax dx' 



d\V) . 



^^3 ' "^c. are = at every point. This is the condition for the re- 

 flection of rays to a focus. 



4. But though the condition V = C and all its consequences are 

 necessary for the convergence of reflected rays to a focus, yet this 

 condition is not necessary for the convergence of a very small pencil 

 of rays incident on the reflecting surface. It is only necessary for 



this, that the equations -^ = 0, and -^^ = should hold at the 



same time, when x = x ^ Ix and V has the corresponding value; that 

 is, that the following equations should be true at the same time, 



dx - 0' 



d{V) d\V) Ix d\V) (Sxf . 



From this we obtain ^^ + ^^.^ + &c. = 0, as the equation 



expressing that the rays incident at the points x and x + Sx intersect : 

 and making Sx indefinitely small, this reduces itself as nearly as we 



please to the equation ^-^ = 0. This tlien is the equation which 



must hold for the ultimate convergence of rays. 



."Jc 2 



