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PROBLEMS AND SOLUTIONS. 
587 
ntioned 
d conic 
ng the 
rsection 
hord is 
the line 
ie given 
md the 
required 
in the plane of the three given points) such that QA + AP = QB + BP = QG + CP = 
given major axis. And this being so, if the locus of P be a given surface, then we 
shall have a certain surface, the locus of Q; and so if the locus of P be a given 
curve in space, then we shall have a given curve in space, the locus of Q. In 
particular, if the locus of P be the plane of the three given points, then the locus 
of Q will be a certain surface, cutting the plane in a curve which is the locus in 
the foregoing problem; and when Q is situate on this curve, then also P will be 
situate on the same curve. Or if the locus of P be the curve in question, then the 
locus of Q will be the same curve. Say, in general, that the loci of P and Q are 
reciprocal loci, then the curve in the problem is its own reciprocal. And we may 
propose the following question: 
Find the curve or surface, the locus of P, which is its own reciprocal. 
We have also analogous to the original problem the following question in Solid 
Geometry : 
Given the four points A, B, C, D in space, to find the locus of the points P, Q 
such that 
PA + AQ = PB + BQ = PC + CQ = PD + DQ = a given line.} 
points, 
A, and 
on the 
y circle 
passes 
H, B. 
cribe a 
ints on 
ellipse 
eral in 
general 
Solution by the Proposer. 
In general if a, b, c be the sides of a triangle, and f, g, h the distances of any 
point from the angles of the triangle (or, what is the same thing, if (a, b, c, f, g, h) 
be the distances of any four points in a plane from each other), then we have a certain 
relation 
</> (a, b, c, f g, h) = 0. 
Hence if r, s, t be the distances of the one focus from the angles of the triangle, and 
the major axis is = 2\, then the distances for the other focus are 2A. — r, 2X — s, 2X — t; 
and considering the three angles and the other focus as a system of four points, we 
have 
cp (a, b, c, 2\ — r, 2\ — s, 2X — t) = 0, 
which is a relation between the distances r, s, t of the first focus from the angles of the 
triangle, and which, treating these distances as coordinates (of course in a generalised 
sense of the term “ Coordinate ”), may be regarded as the equation of the required 
locus. It is to be observed, that we have identically 
cf) (a, b, c, r, s, t) = 0, 
and the equation may be expressed in the simplified form 
(f> (a, b, c, 2\ — r, 2X — s, 2X — t) — cf) (a, b, c, r, s, t) = 0. 
74—2