442 
ON THE DETERMINATION OF THE 
[476 
It is easy to verify that 
Hyperbola 2 passes through x. 2 =—^, y 2 = |-v3, an d touches there circle x 2 + y' 2 = l, 
Vz— 1 
3 
and we thus have the figure in the Plate. 
85. The figure shows the motion of the points 1, 2, 3, along their respective 
hyperbolas, viz. c = 0° to 90°, the point 1 moves from contact with the circle, along a 
half branch to infinity: 2 moves from contact along a small portion of the half 
branch; 3 moves from contact, along the half branch to infinity for c = tan -1 2 = 63° 26', 
and then reappearing at the opposite infinity, as c increases to 90° describes a portion 
of the opposite half branch. 
86. For c — 0, the orbit is the circle; as c increases the orbit becomes elliptic; 
then parabolic, c = 51°, and afterwards hyperbolic (concave); until for c = 60°, the three 
points are on the horizontal line of the figure, and the orbit is this right line; it 
is to be noticed that the arrangement of the points on these orbits is 1, 2, 3; so 
that for the parabola, T 31 is = oo, and for the hyperbolas and right line T 31 does not 
exist. 
87. For c<60° until c = 63° 26' the orbit is a convex hyperbola, the arrangement 
of the points being still 1, 2, 3: say for c = 63° 26' — e, the orbit is the convex 
hyperbola il. At c = 63° 26' there is an abrupt change of orbit; say for c = 63° 26' + e. 
the orbit is a concave hyperbola ; and for c = 65° 52' the orbit is a parabola; 
the arrangement of the points on these orbits is 2, 1, 3; so that for the hyperbolas 
T03 does not exist, and T 23 is = go for the parabola. Observe also that for the 
hyperbola il x , the point 3 is at infinity, or we have T 31 = 00. As c continues to 
increase, the orbit becomes an ellipse, the eccentricity having a minimum value = - 628 
(about), for c = 69° (about). For c = 89° 20' the orbit is again a parabola, and then 
until c = 90° it is a hyperbola; the order of the points on the last-mentioned 
parabola and hyperbolas being 1, 3, 2; so that for the parabola T 12 is = 00, and for 
the hyperbolas T 12 does not exist. In the hyperbola for c = 90°, say the hyperbola O', 
the point 1 is at infinity, or we have T 12 = go . The foregoing results, obtained (except 
as to the numerical values) by consideration of the figure, will be confirmed by means 
of the calculated values of e. 
88. The equation of the orbit may be written 
r 
x 
1 
V3 
V3 
r 2 cos c , 1, sin c , cos c 
r 2 (sin c + 2 cos c), — 1, 2 sin c + 3 cos c, sin c + 2 cos c 
r 3 (sin c — 2 cos c), 1, — 2 sin c + 3 cos c, sin c — 2 cos c 
r 1 COS G 
cos c