121 



rf/3 = -. — J- . da + ■ , , . ao + -T — r- • «c, 



d^V \ 



gives, (when we put rf'a = 0, d^b = 0, rfc = 0, , • = 0, J 



and tjierefore 



My d^y cP V 



rf^/3 dW rf'/3 d^V d^/i d^V 



da* ~ dx'^.dy da.dh ~ dx.dy''^ db* ~ df ' .. , ■ , . 



{V) being the characteristic of the system; so that finally, the coefficients in the formula (N") 

 have for expressions 



d'Y _ ^ dW d'Y _ , £V_ d-Y _ , £V_ 



~da^-^'-^'db^' inh - ^^'^^ ' dxdy-' rf/3' -«'• d^ • ^^ ^ 



They may therefore be calculated, either immediately from the characteristic function (F), if 

 the form of that function be given ; or from the equation of the mirror, and the characteristic of 

 the incident system, according to the method of Section VI. 



[60.] The formula (iV"), which for conciseness may be written thus 



combined with the equation x' = i.«^, in which (?) denotes the interval (j j — gj) between the 

 two foci of the ray, enables us to find the curve in which any thin pencil /3^ =y"(«/)» is cut by a 

 perpendicular plane passing through a focus of the given ray ; a question for which the formulas 

 of Section IX. are not sufficient ; since, by those formulae, the curve would reduce itself to a 

 right line, namely the tangent to the caustic surface. Suppose, for example, that all the rays 

 of the thin pencil make with the given ray some given small angle (^;, in which case we have 

 seen that an ordinary section of the pencil is a little ellipse (M") ; we then have to eliminate 

 a„ /3,, between the three equations 



x'= ix„ y = i(^«/ + 2i?«,A + Cfi,"-), «/ + /3,« = t\ 



and we find for the equation of the section 



2i\y' =: Ax"±2Bif.^(i'S^ — x'^) + C(i^6^ — af*); (P") 



which evidently represents a curve shaped like an hour-glass, or figure of eight, having its node 



on the axis of (y), that is, on the normal to the caustic surface, at a distance = ^Cf^ from the 



focus, and bounded by the two tangents x' = ± iS. The area of this curve, is the double of 



2B 

 the definite integral ^^/y'(j*«» _ x'*). x'dx', taken from x' z:0 to x'=ie ; it is therefore 



s2 



