33 



that is the plane of osculation. This plane becomes a tangent to one 

 of the developable pencils, when the distance R attains either of its 

 extreme values, corresponding to the two points where the ray 

 touches the caustic surfaces, and determined by the equation, 



which results by elimination from the two following : 



(D") 



Let ill , J^2 » be the two values of R, determined by the formula 

 (C-^), and ^1 , <^2 , the two corresponding values of the angle ^, 

 which may be deduced from the following equation : 



then the general relation (J") between jR and 0, may be put under 

 the following form ; 



R-r-Ri /sin.(?.--_^,)y 



^ being a coefficient which is independent of R and ^, and is positive 

 or negative according as the quantity 



is positive or negative. This latter quantity is the same with that 

 which we have before denoted by t;", because the remaining parts of 

 the general expression for v\ namely 



* ^"i' 3/3^ b«S^/ . + 3/3" V "~ ipyj "*" V V "" ^Vj ' 



VOL. XVI. F 



