3NS APPENDIX 



Substituting these values in equation (-J) it becomes 



<> 



and the limit of this equation as a approaches oo,/> remaining constant, is 



y f = 4|; - (0 



which is the equation of a parabola, and the proposition is proved. 



In the same way it may be shown that the parabola is the limit to 

 which an hyperbola approaches when its center moves away to infinity, 

 a vertex and the corresponding focus remaining fixed in position 

 (cf. also Note D). 



NOTE F 



Confocal conies. Two conies having the same foci, F l and F v are 

 called confocal conies. Since the transverse axis of a conic passes through 

 th- foci and its conjugate axis is perpendicular to, and bisects, tin; lino 

 joining the foci, therefore confocal conies are also coaxial,* i.e., they have 

 their axes in the same lines. If the equation of any one of such a system 

 of conies is 



S+f-i, - <D 



and if X is an arbitrary parameter, then the equation 



will represent any conic of the system. For, a and b being constant, and 

 a > 6, equation (2) represents ellipses for all values of X between oo 

 and 6 2 , hyperbolas for all values of X between b 9 and a 2 , and 

 imaginary loci when X < a 2 ; moreover, the distance from the center 

 O to either focus for each of these curves is 



V(a 2 -f X) - (6 a -I- X), 



which equals Va* - ft 2 , and is therefore constant. 



The individual curves of the system represented by equation (3) aro 

 obtained by giving particular values to X, each value of X determining 

 one and but one conic. If any one of these conies is chosen as the 



Coaxial conies are, however, not necessarily confocal. 



