408 
ON THE DETERMINATION OF THE 
[476 
which, attributing therein to r X) r 2 , r 3 , the signs +, — at pleasure, represents eight 
different equations: these however give only four conics, viz., we have the same conic 
whether we attribute to r x , r 2 , r 3 , any particular combination of signs, or reverse all 
the signs simultaneously. 
19. But the focal equation r—Ax + By+C is precisely equivalent to the equation 
1 + e cos (0 — -to-)’ 
and in this equation (taking as is allowable p as positive) then if ± e be = or < 1, 
that is for an ellipse or parabola whatever be the value of 6 — «r, r is always 
positive; but if ± e be >1, that is for a hyperbola, r is positive for those values 
of 0 — -uy which belong to one branch, negative for those which belong to the other 
branch, of the curve. Hence in the determinant equation, unless r Xt r 2 , r 3 , have the 
same sign, the curve will be a hyperbola with the points two of them on one branch, 
the third on the other branch thereof. But in the remaining case, when r x , r 2 , r s , 
have all the same sign, or say when they are all positive, then the conic is an ellipse 
or parabola, or else it is a hyperbola with the three points on the same branch 
thereof; that is, the foregoing determinant equation, regarding therein r x , r 2 , r 3 , as all 
of them positive, gives the orbit. 
20. When one of the points is at infinity on a given line there is a discontinuity 
of orbit. To explain this, suppose that the point (x x , y x ) is situate on the line 
y = x tan cl, at an indefinitely great distance r x in one or the other direction along the 
line; viz., r x is an indefinitely large positive quantity, and we have in the one case 
x x , 2/i = r x cos cl, Tqsma; and in the other case x x , y x — —r x cos a, — Tysina: the corre 
sponding equations of the orbit, putting therein ultimately r x = + oo, are 
r , 
X, 
y> 
1 
= 0, 
r , 
X, 
y> 
1 
1, 
COS CL, 
sin CL, 
0 
1, 
— COS CL, 
— sin CL, 
0 
r 2 , 
x 2 , 
2/2, 
1 
n, 
x 2 , 
2/2, 
1 
r 3 , 
cc 3 , 
2/3, 
1 
r 3 , 
® 3 , 
2/3, 
1 
which equations belong, it is clear, to two distinct conics; or as the point {x x , y x ) 
passes from a positive to a negative infinity along the given line, there is an abrupt 
change of orbit. It is proper to remark that the two orbits are the very same as 
would be obtained by writing x x , y x = r x cos a, ^sina, r x = + oo and r x = — oo in the 
determinant equation: that is, the orbit passes abruptly from one to another of the 
four conics which belong to the position (x x , y x ), and we thus understand how the 
transition from + oo to — oo, which is geometrically no breach of continuity, occasions 
in the actual problem a discontinuity. 
21. The same thing appears from the geometrical construction; and we derive a 
further result which will be useful. Suppose first that the point 1 is at infinity in 
the direction shown by the arrow; then drawing 2c = 2S and 3b = 3S each in the 
direction opposite to Si, we have the points b, c on the directrix, which is thus the