ENVELOPS. 363 



from which it follows by (6) that 



M + iQ+rQ-o (8). 



do do dc 



Hence we have to eliminate a, b, c, X and fj, from equations 

 (1), (2), (7) and (8) ; the result is the equation to the envelop 

 required. 



Example. A straight line moves so that the length inter- 

 cepted between the co-ordinate axes is constant : required the 

 envelop of the moving straight line. 



Let the equation to the straight line be 



a o 

 so that a 2 +b s = a, constant = fc 2 , say (10). 



From (9) 

 from (10) aDa + bDb = ; 



thus 



therefore -* + Xa = 0, andf^ + X&=0 ............... (11); 



Cii [) 



multiply the first of these equations by a and the second by 

 b and add ; thus 



that is, 1 + \k* = 0, therefore X. = ^ . 



Then from (11) 



Therefore by (9) 



/W fj I 



x -L. y = i 



x.\ 



or x + y = . 



This equation determines the envelop. 



