26 BARKER : 



From which the loci are evidently a pair of ellipses, 

 centres at (D, o) and (— D, o), semi-axes 



dJ(3 + V5) and dJUEjZS 



VI. The equation of the envelope is got without any dif- 

 ficulty by eliminating a between the equation (i) of the 

 family, and the partial derivative of that expression with 

 respect to a. 



Equation ( i ) may be put 



x^sec'^a 



y — xtana -j = o 



4D 



Taking the partial derivative with respect to a, 



„ x^sec^atana 

 — xsec a -I- ^ o 



2D 



Dividing through by xsec'a, 

 xtana 



2D 



2D ., 4 D'' 



Whence tana = - and seca = i -]- 



X x^ 



Substituting in ( r ) , 



y = 2D "" 



4D 



or x' = — 4D (y — D) 



and this is the equation of a parabola with vertical axis, 

 vertex at (o, D), focus at (o, o), and latus rectum 4D. 



