24. 



equations (B') contain the whole theory- of the developable pencils 

 and of the caustic curves and surfaces. Putting them under the 

 form, 



we find by eliminating the differentials, and attending to the rela- 

 tions (G), (2/), the following quadratic equation 



= j'F" — jS + v", {D') 



which may also be thus transformed, 



= i'v" — {S' + W" : (E') 



the symbols if', V", W", S, S', having the same meanings as in the 

 fifth number. The form (DO, serves to connect the distance § with 

 the function F, and the form (E') with W. By either of these forms, 

 we obtain in general two values of §, and therefore two points a/' y" z'\ 

 which are the only points wherein the ray can touch a caustic curve : 

 and the locus of the points thus obtained, composes the two caustic 

 surfaces. The joint equation of these surfaces, in x" y" z", may be 

 found by eliminating a, /3, y, between the four following equations: 



x" = x' + X [»x" + fiy" + yz") , -\ 



y" z=y' + fi(ux" + fiy" + y«"), / 



r" = z' +y (»x" + fiy" + yz"), r ''^ ^ 



= («x" + fiy" + y«")" v" + («x" -f- fiy" + yz") S/ -f WJ' ; ) 



in which S/, W", are formed from S', W", by changing x, y, z, to 



•i3»afill' hint,.' n^mm. dJ^^-iill: tij, .. 



