64 
ON A LOCUS IN RELATION TO THE TRIANGLE. 
[394 
to the Absolute, the point so obtained being a point on the line G'O. To each position 
of 0 on the conic locus, there corresponds of course a position of 0; the locus of C 
is, as has been shown, a quartic curve having a node at each of the points G', A, B. 
The foregoing conclusions apply of course to spherical figures; we see therefore that 
on the sphere the locus of a point such that the perpendiculars let fall on the sides of a 
given spherical triangle have their feet in a line (great circle), is a spherical cubic. If, 
however, the spherical triangle is such that the angles thereof and the poles of the sides 
(or, what is the same thing, the angles of the polar triangle) lie on a spherical conic ; 
then the cubic locus breaks up into a line (great circle), which is in fact the circle 
having for its pole the point of intersection of the perpendiculars from the angles of the 
triangle on the opposite sides respectively, and into the before-mentioned spherical conic. 
Assuming that the angles A and B are given, the above-mentioned construction, by 
means of the point 0, is applicable to the determination of the locus of the remaining 
angle G, in order that the spherical triangle ABC may be such that the angles and 
the poles of the sides lie on the same spherical conic, but this requires some further 
developments. The lines G'B', G'A' which are the polars of the given angles A, B 
respectively, are the cyclic arcs of the conic the locus of 0, or say for shortness the 
conic 0; and moreover these same lines G'B', G'A' are in regard to the conic 0, 
the polars of the angles B, A respectively. If instead of the conic 0 we consider the 
polar conic O', it follows that A, B are the foci, and G'A', G'B' the corresponding 
directrices of the conic O'. The distance of the directrix G'A' from the centre of the 
conic, measuring such distance along the transverse axis is clearly = 90° — distance of 
the focus A ; it follows that the transverse semi-axis is = 45°, or what is the same 
thing, that the transverse axis is = 90; that is, the conic 0' is a conic described 
about the foci A, B with a transverse axis (or sum or difference of the focal distances) 
= 90\ Considering any tangent whatever of this conic, the pole of the tangent is a 
position of the point 0, which is the point of intersection of the perpendiculars let 
fall from the angles of the spherical triangle on the opposite sides; hence, to complete 
the construction, we have only through A and B respectively to draw lines AG, BG 
perpendicular to the lines BO, GO respectively; the lines in question will meet in a 
point G, which is such that GO will be perpendicular to AB, and which point C is 
the required third angle of the spherical triangle ABC. In order to ascertain whether 
a given spherical triangle ABC has the property in question (viz. whether it is such 
that the angles thereof and of the polar triangle lie in a spherical conic), we have 
only to construct as before the conic 0' with the foci A, B and transverse axis = 90’’, 
and then ascertain whether the polar of the point 0, the intersections of the perpen 
diculars from the angles of the triangle on the opposite sides respectively, is a tangent 
of the conic O'. It is moreover clear, that given a triangle ABC having the property 
in question, if with the foci A, B and transverse axis = 90 we describe a conic, and 
if in like manner with the foci A, G and the same transverse axis, and with the foci 
B, G and the same transverse axis, we describe two other conics; then that the three 
conics will have a common tangent the pole whereof will be the point of inter 
section of the perpendiculars from the angles of the triangle ABC on the opposite sides 
respectively.