_u_ and _u’_ are in perspective position, and that lines through
corresponding points all pass through a point _M_, the center of
perspectivity, the position of which will be determined by any two such
lines. But the intersection of _a_ with _u_ and the intersection of _c’_
with _u’_ are corresponding points on _u_ and _u’_, and the line joining
them is clearly _c_ itself. Similarly, _b’_ joins two corresponding points
on _u_ and _u’_, and so the center _M_ of perspectivity of _u_ and _u’_ is
the intersection of _c_ and _b’_. To find _d’_ in _S’_ corresponding to a
given line _d_ of _S_ we note the point _L_ where _d_ meets _u_. Join _L_
to _M_ and get the point _N_ where this line meets _u’_. _L_ and _N_ are
corresponding points on _u_ and _u’_, and _d’_ must therefore pass through
_N_. The intersection _P_ of _d_ and _d’_ is thus another point on the
locus. In the same manner any number of other points may be obtained.




*65.* The lines _u_ and _u’_ might have been drawn in any direction
through _A_ (avoiding, of course, the line _a_ for _u_ and the line _a’_
for _u’_), and the center of perspectivity _M_ would be easily obtainable;
but the above construction furnishes a simple and instructive figure. An
equally simple one is obtained by taking _a’_ for _u_ and _a_ for _u’_.




*66. Lines joining four points of the locus to a fifth.* Suppose that the
points _S_, _S’_, _B_, _C_, and _D_ are fixed, and that four points, _A_,
_A__1_, _A__2_, and _A__3_, are taken on the locus at the intersection
with it of any four harmonic rays through _B_. These four harmonic rays