we hare 



95 



(PF , (PV ePF 



da'' '^^'daM'^ ^' da.dc ~ °' 

 d^V d^V , dW 



daM ^ db- ' ' db.dc 



(f F d^V d^V _ 



*■ d^ + '^' dli.dc "*■ ^" rf? ~ °' 



"*' da" db^ \da.db) ' 



we find this other form for the equation of the caustic surfaces, 



1 id«r d^r /d^vy-} ^, cd'V,d'V d^Vf 



v[d^- -d¥-{iZdb) 5-« + irf? + ^ + i?5-« + ^ = ^' W 



[28.] The manner in which these formulae are to be employed is evident. We are to inte- 

 grate (P) considered as a differential equation, of the first order and second degree, between 

 », /3, or between the corresponding functions of x, y, z. 



the integral will be of the form 



dV _ dV 



dx ^ ~ dy ' 



=/r^.^) 



dy 



C being an arbitrary constant ; the condition of passing" through a given ray will determine 

 the two values of this constant, corresponding to the two developable pencils : and the equations 

 of the caustic curves, considered as the aretes de rehroussement of those pencils, will follow by 

 the known methods from the equations of the pencils themselves. The points in which a given 

 ray touches these caustic curves, that is the two foci of the ray, are determined, without any inte- 

 gration, by means of (Q) or (R) ; and thus we can determine, by elimination alone, the equations 

 of the two caustic surfaces, the locus of those points or foci. 



[29.] In the preceding reasonings, we have supposed given the form of the characteristic 

 function V, whose partial differential coefficients of the first order, are equal to the cosines of 

 the angles that the reflected ray makes with the axes ; let us now see how the partial differential 

 coefficients of the second order of this function, which enter into the formulae that we have found 

 for the developable pencils and for the caustic surfaces, depend on the curvature of the mirror, 

 and on the characteristic function of the incident system. Let ( V) represent this latter func- 

 tion, so that we shall have 



, dV , dV dV 



* -~d^' ^-~d^' '^-'dT' 



VOL. XV. P 



