BOOK OF MATHEMATICAL PROBLEMS. 

 2 



= r^sin-0' 



a pair of parabolas having the given point for focus, and their 

 common latua rectum in the given direction and of twice the 

 given length. This admits of very easy geometrical proof. 



In this case every one of the system of curves has real 

 contact with the envelope; but it frequently happens that this 

 is true for only a portion of the system. 



The same principle applies to systems whose equation involves 

 higher powers of the parameter ; but in the following examples 

 it will be found that the general equation of the system whose 

 envelope is required can always be expressed in the form 

 U+2\V+\*W=Q, where X is the parameter; and the cor- 

 responding envelope is the curve UW= V*. 



729. The envelope of the circles 



wl.ere /?, y are connected by the equation 



1 1_1 



/r r -a' 



is the two circles a?' + y 9 2ax = 0. 



730. The envelope of the circles 



where ft, y are connected by the equation 

 (P + a)(y-a) + h a =0, 

 is the two circles y* + (x 7t)' = a*. 



731. The ellipse ^ + ^ =1 



has its- axes connected by the equation 



