*47. Superposed fundamental forms. Self-corresponding elements.* We have seen (§ 37) that two projective point-rows may be superposed upon the same straight line. This happens, for example, when two pencils which are projective to each other are cut across by a straight line. It is also possible for two projective pencils to have the same center. This happens, for example, when two projective point-rows are projected to the same point. Similarly, two projective axial pencils may have the same axis. We examine now the possibility of two forms related in this way, having an element or elements that correspond to themselves. We have seen, indeed, that if _B_ and _D_ are harmonic conjugates with respect to _A_ and _C_, then the point-row described by _B_ is projective to the point-row described by _D_, and that _A_ and _C_ are self-corresponding points. Consider more generally the case of two pencils perspective to each other with axis of perspectivity _u’_ (Fig. 9). Cut across them by a line _u_. We get thus two projective point-rows superposed on the same line _u_, and a moment’s reflection serves to show that the point _N_ of intersection _u_ and _u’_ corresponds to itself in the two point-rows. Also, the point _M_, where _u_ intersects the line joining the centers of the two pencils, is seen to correspond to itself. It is thus possible for two projective point-rows, superposed upon the same line, to have two self-corresponding points. Clearly _M_ and _N_ may fall together if the line joining the centers of the pencils happens to pass through the point of intersection of the lines _u_ and _u’_. [Figure 10] FIG. 10 *48.* We may also give an illustration of a case where two superposed projective point-rows have no self-corresponding points at all. Thus we may take two lines revolving about a fixed point _S_ and always making the same angle a with each other (Fig. 10). They will cut out on any line _u_ in the plane two point-rows which are easily seen to be projective. For, given any four rays _SP_ which are harmonic, the four corresponding rays _SP’_ must also be harmonic, since they make the same angles with each other. Four harmonic points _P_ correspond, therefore, to four harmonic points _P’_. It is clear, however, that no point _P_ can coincide with its corresponding point _P’_, for in that case the lines _PS_ and _P’S_ would coincide, which is impossible if the angle between them is to be constant. *49. Fundamental theorem. Postulate of continuity.* We have thus shown that two projective point-rows, superposed one on the other, may have two points, one point, or no point at all corresponding to themselves. We proceed to show that