ON THE DETERMINATION OF THE 
410 
[476 
= — r 2 cos a 2 , — r 2 sina 2 , where n is an indefinitely large positive quantity. And writing 
ultimately r x = + oo, r. 2 = + co, the equation of the orbit is obtained in the form 
r, 
X , 
y » 
1 
1, 
+ cos oq, 
+ sin Otj, 
0 
1, 
+ cos a 2 , 
+ sin a 2 , 
0 
r 3 , 
^3 , 
2/3 , 
1 
where the + of the second line and the + of the third line have each of them the 
value + or — at pleasure. There are consequently four distinct orbits, corresponding 
to the combinations of each of the two directions of the point 1 with each of the 
two directions of the point 2. And it is moreover clear that these are the very conics 
which are obtained from the determinant equation by writing therein x 1} y 1 — r x cos a 2 , 
rjsinaj; x. 2 , y 2 = r 2 cos a 2 , ? , 2 sina 2 and r 2 = + qo , —oo; r 2 = +cc, —oo successively; viz., 
the orbit changes abruptly between the four conics which correspond to the given 
position of the points 1, 2, 3. 
Fig. 4. 
26. The geometrical construction is very simple indeed; viz., measuring off from 
3 in the directions $ 1, S 2, and in the opposite directions respectively, a distance 
= S3, we have four points, the angles of a rectangle; and joining these in pairs, we 
have the four positions of the directrix: the figure shows at once that the orbits are 
three of them concave, the remaining one convex. 
27. The determinant equation obtained for the orbit is an equation of the form 
r = Ax + By + C; 
and it is clear that the equation of the directrix is Ax + By + 0 = 0. By what 
precedes, this line will lie on the same side of the three points; viz., either it does 
not separate them from the focus, and the orbit is then concave, or it does separate 
them from the focus, and the orbit is then convex. Although in general the sign of 
G is no criterion (for the equations r = Ax + By + C and r — — Ax—By — G represent