16 DR. J. BOTTOMLEY ON COMPOUND 



equations (3) and (4) ; but it is evident that by means of 

 the same equations we may solve inverse problems, viz. 

 given the equation to the projectrix to deduce that of the 

 primitive. If the equation to the projectrix be given in 

 the form 



that of the primitive will be of the form 



f\x,m{l{w—a)-\-m[y — b))j. 



Suppose the projectrix to be the circle 

 y^ -{- {x — ay =■ c^ , 

 we shall obtain for the primitive the ellipse 



{x—ay{i +m^l^) +2tn}l{x — a){y — b) +m*{y — b)^ — c^. 

 The semiaxes of this ellipse are 



^ C^/2 



and 



C\/2 



Although the projection of this ellipse on the axis of x 

 may be a circle, its projection on the axis of y will not 

 simultaneously be a circle. The projectrix in this case 



will be 



{x-lm{y-b)y + l^m^{y-by = l^c^, 



representing an ellipse of which the semiaxes are 



cZV2 



\/ 1 + 2m^r + x/^.l'-m'- + I 

 and 



\/ 1 + 2/w^r — v'4m^/^ + I 



