On Systems of Rays. 77 
30’. cos. #' =< .2l. cos. ’3 
a (Z”") 
6’. sin. ¢ =~—.&l. sin. ¢ : 
while the equations (JV”*) of the initial ter reduce themselves to the following, 
{oo aA oa’ a ipa BAN 0B" ll 
&L=2 Se Or 3 Y =e By. oY (A ) 
If, then, these initial ray-lines form a circular cone haying for equation 
wet y? = 280", (B") 
the corresponding locus of the final point B,, on the final plane of xy, will not in 
general be a circle, but an ellipse, having for its equation 
(Sy aa + (Py. Sy? = 30°, (C") 
of which, by ( Y"), the axis of a coincides with the least and the axis of y with the 
greatest axis ; and reciprocally if the final locus be a circle having for equation 
82° + oy = 81’, (D") 
the initial cone of ray-lines will have for equation 
e@ aS ae, ay 
so that its perpendicular sections are ellipses, having their greater axes in the plane of 
x 2, and their lesser axes in the plane of y/ 2’. It is evident that a circle equal to 
the final circle (D") may be obtained from the elliptic cone (#"), by cutting that 
elliptic cone by any one of the four following planes, 
; Sa \— e\ 
dae (E) ey (ES) (GF) 9 
and in like manner the four elliptic sections of the circular cone (B"), made by the 
same four planes, are all equal and similar to the final ellipse (C'"). In general it is 
easy to prove by the equations of the initial ray-lines (4"), that whatever final locus 
we take for the point B,, represented by the equation 
ey =f (6x); (G") 
the corresponding initial cone 
VOL. XVII. x 
