PROJECTION 795 and then ST = SJ + SF or SI = JT, SJ = I T. In the second case (fig. 4) i andy will be on opposite sides of the axis, and the centre S will lie between them in such a position that I S = TJ and I / T = SJ. If I S = SJ, the point S will lie on the axis. It follows that any one of the four points S, T, J, F is completely determined by the other three : if the axis, the centre, and one of the lines i or j are given the other is determined ; the three lines s, i , j determine the centre ; the centre and the lines i , j deter mine the axis. 9. We shall now suppose that the two projective planes IT, IT are perspective and have been made to coincide. THEOREM. If the centre, the axis, and either one pair of corre sponding points on a line, through the centre or one pair of corre sponding lines meeting on the axis are given, then the whole projection is determined. Proof. If A and A (fig. 1) are given corresponding points, it has to be shown that we can find to every other point B the cor responding point B . Join AB to cut the axis in R. Join RA ; then B must lie on this line. But it must also lie on the line SB. Where both meet is B . That the figures thus obtained are really projective can be seen by aid of Desargue s theorem. For, if for any point C the corresponding point G be found, then the triangles ABC and A B C are, by construction, co-linear, hence co-axal ; and s will be the axis, because AB and AC meet their corresponding linns A B and A C on it. BC and B C therefore also meet on s. If on the other hand a, a are given corresponding lines, then any line through S will cut them in corresponding points A, A which may be used as above. 10. Rows and pencils which are projective or perspective have been considered in the article GEOMETRY (G. 12-40). All that has been said there holds, of course, here for any pair of correspond ing rows or pencils. The centre of perspective for any pair of corresponding rows is at the centre of projection S, whilst the axis contains coincident corresponding elements. Corresponding pencils on the other hand have their axis of perspective on the axis of projection whilst the coincident rays pass through the centre. We mention here a few of those properties which are indepen dent of the perspective position : The correspondence between two projective roivs, or pencils, is com pletely determined if to three elements in one the corresponding ones in the other are given. If for instance in two projective rows three pairs of corresponding points are given, then we can find to every other point in either the corresponding point (G. 29-36). If A, B, C, D are four points in a row and A , B , C , D the corre sponding points, then their cross-ratios are equal (ABCD) = ( A B C D ), where (ABCD) = AC/CB : AD/DB. If in particular the point D lies at infinity we have (ABCD) = - AC/CB = AC/BC. If therefore the points D and D are both at infinity we have AC/BC = AD/BD, and the rows are similar (G. 39). This can only happen in special cases. For the line joining corresponding points passes through the centre ; the latter must therefore lie at infinity if D, D are different points at infinity. But if D and D coincide they must lie on the axis, that is, at the point at infinity of the axis unless the axis is altogether at infinity. Hence In two perspective plants every row w-hich is parallel to the axis is similar to its corresponding row, and in general no other row has this property. But if the centre or tJie axis is at infinity then every row is similar to its corrcs2)onding row. In either of these two cases the metrical properties are particu larly simple. If the axis is at infinity the ratio of similitude is the same for all rows and the figures are similar. If the centre is at infinity we get parallel projection ; and the ratio of similitude changes from row to row (see 16, 17). In both cases the mid-points of corresponding segments will lie corresponding points. _ 11. INVOLUTION. If the planes of two projective figures coin cide, then every point in their common plane has to be counted twice, once as a point A in the figure IT, once as a point B in the figure -IT . The points A and B corresponding to them will in general be different points ; but it may happen that they coincide. Here a theorem holds similar to that about rows (G. 76 sq.). THEOREM. If tivo projective planes coincide, and if it happens that to one point in their common plane the same point corresponds, whether we consider the point as belonging to the first or to the second plane, then the same will happen for every other point that is to say, to every point will correspond the same point in the first as in the second plane . In this case the figures are said to be in involution. Proof. Let (fig. 5) S be the centre, s the axis of projection, and let a point which has the name A in the first plane and B in the second have the property that the points A and B corresponding to them again coincide. Let C and D be the names which some other point has 8 / ^N/V"^ F* s in the two planes. If the line AC cuts the axis in X, then the point where the line XA cuts SC will be the point C corresponding to C ( 9). The line B D also cuts the axis in X, and therefore the point D corresponding to D is the Fig. 5. point where XB cuts SD . But this is the same point as C . Q.E.I). This point C might also be got by drawing CB and joining its intersection Y with the axis to B . Then C must be the point where B Y meets SC. This figure, which now forms a complete quadrilateral, shows that in order to get involution the correspond ing points A and A have to be harmonic conjugates with regard to S and the point T where AA cuts the axis. THEOREM. If two perspective figures are in involution, two cor responding points are liarmonic conjugates with regard to the centre and the point where the line joining them cuts the axis. Similarly Any two corresponding lines are harmonic conjugates with regard to the axis and the line from their point of intersection to the centre. Conversely If in two perspective planes one pair of corresponding points are harmonic conjugates with regard to the centre and the point where the line joining them cuts the axis, then every pair of corresponding points has this property and the planes are in involu tion. 12. PROJECTIVE FLAXES WHICH ARE NOT IN PERSPECTIVE POSITION. We return to the case that two planes IT and IT are pro jective but not in perspective position, and state in some of the more important cases the conditions which determine the correspondence between them. Here it is of great advantage to start with another definition which, though at first it may seem to be of far greater generality, is in reality equivalent to the one given before. DEFINITION. We call two planes projective if to every point in one corresponds a point in the other, to every line a line, and to a point in a line a point in the corresponding line, in such a manner (hat the cross-ratio of four points in a line, or of four rays in a pencil, is equal to the cross-ratio of the corresponding points or rays. The last part about the equality of cross-ratios can be proved to be a consequence of the first. But as space does not _ allow us to give an exact proof for this we include it in the definition. If one plane is actually projected to another we get a correspond ence which has the properties required in the new definition. This shows that a correspondence between two planes conform to this definition is possible. That it is also definite we have to show. It follows at once that Corresponding rows, and likewise corresponding pencils, are pro jective in the old sense (G. 25, 30). Further, If two planes are projective to a third they are projectile to each other. THEOREM. The correspondence between two projective planes IT and IT is determined if we have given either two rows u, v in TT and the corresponding rows u , v in IT , the point where u and v meet corre sponding to tlie points where u and v meet, or two pencils U, V in w and the corresponding pencils U , V in IT , the ray IJV joining the centres of the pencils in TT corresponding to the ray U V. It is sufficient to prove the first part. Let any line a cut it, v in the points A and B. To these will correspond points A and B in u and v which are known. To the line a corresponds then the line A B . Thus to every line in the one plane the corresponding line in the other can be found, hence also to every point the corre sponding point. 13. THEOREM. If tlie planes of two projective figures coincide, and if either four points, of which no three lie in a line, or else f out lines, of which no three pass through a point, in the one coincide with their corresponding points, or lines, in the other, then every point and every line coincides with its corresponding point or line so that the figures are identical. If the four points A, B, C, D coincide with their corresponding points, then every line joining two of these points will coincide with
its corresponding line. Thus the lines AB and CD, and therefore also