t£ 





being changed to their expressions (P), or rather to the equivalent 

 expressions, 



i' ^£ = M,J* + m; J/3 4- P;jy + («a,- + /s^ + ^a) s |i , J 



^' & = ^^''^* + ^'^^ + -^'■'^y + («X + /3y + yz) § ^ , V (N') 



^' ^ = ■P''^* -f- ^'^VS^ + P,h + i"^ + /3y + yz) J 1^ , j 



from which ax 4-/31/4-72; M'ill disappear, when substituted in the 

 equations (M'), and in which 



•"> + '^^ + "-f =. ^ f - ( '-' S + '■' w + '•' £)• [ '°'> 



10. A remarkable transformation of the equations (B), which 

 determine, as we have seen, the developable pencils, and the caustic 

 curves and surfaces, may be obtained in the following manner. We 

 have by (P), 



which gives 



when we substitute for x, y, z, their expressions of' 4- a^, y-' 4- /3^, 

 2^' + yf , and attend to the relations (G). And by similar substitu- 

 tions in the expressions for 



b2 



