Full text: The collected mathematical papers of Arthur Cayley, Sc.D., F.R.S., sadlerian professor of pure mathematics in the University of Cambridge (Vol. 7)

456 
ON THE DETERMINATION OF THE 
[476 
or, what is the same thing, 
xi = cosec b 
2// = V 3 cot b, 
sin 6 — V3 cos b ’ 
or as these may also be written 
sin b + V3 cos 5 ’ 
, _ V3 (cos 5 — V3 sin 5) 
sin 6 + V3 cos b 
, V3 (cos b + V3 sin 5) _ 
sin 6 — V 3 cos 5 
aV = cosec Z> , yí = V3 cot 6 , 
= — cosec (& + 60°), y¿ — V3 cot (6 + 60°), 
a?/ = — cosec (6 — 60°), y¿ = V3 cot (6 — 60°), 
so that for each of these sets we have 
(The curve is in fact a section of the hyperboloid of revolution, a; 2 + y 2 — = 1, 
which passes through the three rays.) 
101. As regards the equation of the orbit I will first consider the particular 
cases 5 = 90°, b = 0°, which should agree with the orbits for c = 90° in the planograms 
1 and 2 respectively. 
For b = 90° we have x = x, y' — y and 
and the orbit is at once found to be 
r = J(l-Vl3)(«'-l), 
the eccentricity (regarded as positive) being thus ^ (v^lS — 1), ='7685 as before. For 
b = 0° there is a discontinuity, and I write successively b = +e, and b = — e. For b = + e 
we have x' = — y, y' = x, and 
Xi = oo , y{ — oo V3,
	        
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