426 
ON THE DETERMINATION OF THE 
[476 
colatitude c. The series of curves in question may be called “graduation curves,” viz., 
they would serve for the graduation of the hyperbola in the planogram for an orbit- 
plane rotating round any line x cos b + y sin b = 0 in the plane of xy. But the elimination 
cannot be easily effected, and I am not in possession of any method of tracing the 
series of curves. 
60. I remark that from the equations 
x' : y' : 1 = (a cos b + b sin b) cos c — c sin c 
: — a sin b + b cos b 
: (f cos b + g sin b) sin c + h cos c, 
we may without difficulty eliminate b; the result is, in fact, 
[?) (— ah cos c) + y (— bh cos' 2 c — eg sin 2 c) — ac sin c] 2 
4- \x ( bh cos c) + y' (— ah cos 2 c -1- cf sin 2 c) + be sin c] 2 
= [x' ( ch sin c) + y' ( ag — bf) sin c cos c + (a 2 — b 2 ) cos cf, 
a conic; but the geometrical signification of this result is not obvious, and I do not 
make any use of it. 
Article Nos. 61 to 63. The Trivector and the Orbit 
61. Considering now the three rays, these are determined by their six coordinates, 
( a i> h l5 c i> fi> hi), 
(a 2 , b 2 , c 2 , f 2 , g 2 , h 2 ), 
(a 3 , b 3 , c 3 , f 3 , g 3 , h 3 ), 
respectively; and the intersections with the orbit-plane are given by 
xi : y r ' : 1 = (a l5 b 15 c^a, /3, 7) : -(a 1; b 1} c&a.', /37) : (f 1; g lt h^a", /3", 7"), 
xi : yi : 1 = (a 2 , b 2 , c a $ „ ) : - (a 2 , b 2 , c 3 # „ ) : (f 2 , g a , h a ][ „ ), 
x-1 : yi : 1 = (a*, b 3 , c 3 $ „ ) : - (a 3 , b 3 , c 3 ][ „ ) : (f 3 , g s , h 3 ][ „ ), 
where the axes Sx', Sy, are an arbitrary set of rectangular axes in the orbit-plane; 
or where, as before, the axis Sx' may be taken to be the line of nodes. 
There is no difficulty in finding the equation of the orbit. Writing r 1 =^x-i 2 + yf, 
we have 
_ u x 
(f„ g„ h/3". 7")’ 
if 
Ui = ± V[(a x , bj, Cj][a', /3', 7')] 2 +[( a if y)P>