*62. Determination of the locus.* We now show that _it is possible to assign arbitrarily the position of three points, __A__, __B__, and __C__, on the locus (besides the points __S__ and __S’__); but, these three points being chosen, the locus is completely determined._ *63.* This statement is equivalent to the following: _Given three pairs of corresponding rays in two projective pencils, it is possible to find a ray of one which corresponds to any ray of the other._ *64.* We proceed, then, to the solution of the fundamental PROBLEM: _Given three pairs of rays, __aa’__, __bb’__, and __cc’__, of two protective pencils, __S__ and __S’__, to find the ray __d’__ of __S’__ which corresponds to any ray __d__ of __S__._ [Figure 12] FIG. 12 Call _A_ the intersection of _aa’_, _B_ the intersection of _bb’_, and _C_ the intersection of _cc’_ (Fig. 12). Join _AB_ by the line _u_, and _AC_ by the line _u’_. Consider _u_ as a point-row perspective to _S_, and _u’_ as a point-row perspective to _S’_. _u_ and _u’_ are projectively related to each other, since _S_ and _S’_ are, by hypothesis, so related. But their point of intersection _A_ is a self-corresponding point, since _a_ and _a’_ were supposed to be corresponding rays. It follows (§ 52) that _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’_.