94 



-— :- . da + -; — rr • "" + ~; — 7— • dc = da, 

 da^ ^ da.db ^ dauic 



d^V , , dJ'V ,, d^V , , 



dadb db^ dbdc 



and therefore 



.. , t d:'y «^^\ J / dW d^V \ ^^ 



M.da=[^.-^-fi.^^). d. + [fi.^^^-y.^^). dfi, 



^'^^ = V'dbTc-''-d^b}' '^''■^V--d^-''-lUc)-^^' 



if we put for abridgment 



^ £V_ d^V _ d'V d^V l 

 ~ ' (_da.db ' db.dc da.dc ' db^ 5 



■*" ^' idoM ' rfo^ db.dc ' da" 5 



\da.dbj 5 



+ ''^ ^ da" ' db^ 



This being laid down, let (x, y, z) be any other point upon the ray, at a distance ({) from the 

 given perpendicular surface ; we shall have 



rf{ = «,dx -|- /3a[y -J- ydz, 

 dx — «rfg =: rfa + {£?«, c/y — ,Srfj = rf6 + (d^ : 



and if the coordinates (x, y, z) belong to a caustic curve, the first members of these two last 

 equations will vanish, so that on this hypothesis, 



da + {.rf* = 0, db-\- {.rf/3 =r ; 

 eliminating (j) we find, for the developable pencils, 



da. d^—db. d» = 0, (P) 



and eliminating -j— we find, for the caustic surfaces, 

 d» 



(da\ f . db\ da db „ ,^. 



substituting for — -- , —7- , j- > -r; , their values, and observing that by the general relation 

 d» a/3 a* a^ 



/rfF\« /dV\t fdVV , 



