428 
ON THE DETERMINATION OF THE 
[476 
boloid which contains the three rays. And we see that the equation of this reciprocal 
cone is 
(fi, gi, hja", /3", Y"), a i > b,, Cl 
(f a , ga, h 2 ][ „ ), a 2 , b 2 ,c 2 
(is? g3> h 3 ]£ „ ), a 3 , b 3 , c 3 
Article Nos. 64 and 65. The Special Symmetrical System of three Rays. 
64. In what follows I consider the three rays forming a symmetrical system as 
already referred to: viz. the three rays intersect the plane of the ecliptic at points 
equidistant from S at longitudes 0°, 120°, 240°; each of them is at right angles to 
Fig. 6. 
the line joining S with the intersection with the plane of the ecliptic, and at an 
inclination = 60° to this plane: the figure shows the projection on the plane of the 
ecliptic of the portions which lie above this plane of the three rays respectively. 
The three rays lie on a hyperboloid of revolution having the line Sz for its axis; 
the circumscribed or asymptotic cone vertex S, is a right cone of the semi-aperture 
= 30°; the reciprocal cone is therefore a right cone semi-aperture 60°, or (what is the 
same thing) the regulator is a small circle, angular radius 60°, and the regulator and 
separators have the positions shown in fig. 1, see No. 8. 
Taking $1=$2 = $3 = 1, and writing down the equations of the three rays in the 
forms 
x — 1 
0 
u. 
i 
tan 60° ’ 
z 
x + cos 60° _ y — sin 60° _ ^ 
— sin 60° — cos 60° tan 60° ’ 
x + cos 60° _y + sin 60° _ z 
sin 60° — cos 60° tan 60° ’