CORRESI’O1'DE.TCE.] corre.=:pondence between the points in the two rows possessing the first two properties. Such a correspondence has been called a one-one corrrspomlencc, whilst the two rows between which such correspond- ence has been established are said to be projective or homologous. ’l‘wo rows which are each the projection of the other are therefore projective. Vc shall presently see, also, that any two projective rows may always be placed in such a position that one appears as the projection of the other. If they are in such a position the rows are said to be in pm-spect1're position, or simply to be perspective. § 26. 'l‘he notion of a one-one correspondence between rows may lie: extended to llat and axial pencils, viz., a flat pencil will be said to be projective to a llat pencil if to each ray in the first corresponds one my in the second, and if the eross-ratio of four rays in one cllllilln‘ that of the corresponding rays in the second. Similarly an axial pencil may be projective to an axial pencil. lut a Hat pencil may also be projective to an axial pencil, or either peneil may be projective to a row. The definition is the same in each ease: there is a one-one correspoiideiice between the elements, and four elements have the same cross-ratio as the correspondiiig ones. § 27. ’l‘liere is also in each case a special position which is called pr‘/‘spa’/.z"ro, viz. :— 1. Two projective rows are perspective if they lie in the same plane, _uid if the one row is a projection of the other. '3. Two projective flat pencils arc pcrspective——(a) if they lie in the same plane, and have a row as a common section ; (B) if they lie in the same pencil (in space), and are both sections of the same axial pencil ; (-y) if they are in space and have a row as comnion_ section, or are both sections of the same axial peneil, one of the conditions involving the other. Two projective axial pencils, if their axes meet, and if they have a flat peneil as a coiiimoii section. 4. A roiv and a projective flat. pencil, if the row is a section of the pencil, each point lying in its corresponding line. 5. A row and a projective axial pencil, if the row is a section of the pencil, each point lying in its corresponding line. 6. A llat and a projective axial peneil, if the formei‘ is a section of the other, each ray lying in its corresponding plane. That in (3{1I‘ll case the correspoiidciice established by the position indicated is such as has been called projective follows at once from the IlOlllllllOll. It is not so evident that the perspective position may always be obtained. Ve shall show in § 30 this for the iirst three cases. First, however, we shall give a few theorems which relate to the general correspondence, not to the perspective position. § :28. T|IEUl{I£.[.——Tw0 rows or pencils, fiat or axial, which are p/'r)_jr7r_'tirc to (6 third are pro_jecti7:e to each other, as follows at once .'rom the dclinitions. § ‘J9. l"u.'i».uinxTAL TlII£ORE‘.l.—If tu-o rozcs, or taco pencils, c‘1'tlm'_flut or axial, or (6 row and a pencil, shall be projecti'7:c, its
- .-my assume to an y three elements in the one the three corresponding
¢'lv‘)1lI.‘Il[8 in the other, and th.cn the correspondence is mziqucly cvcternzinril. I’roof.—lf in two projective rows we assume that the points A, B. C in the first correspond to the given points A’, B’, C’ in the second, then to any fourth point D in the first will correspond a point D’ in the second, so that (A BCD) = (.-’B’C’D’). But there is only one point, I)’, which makes the cross-ratio (A’B’C’D’) equal to the given number (ABCD). The same reasoning holds in the other cases. § 30. Thcorcm.—lt' two rows are perspective, then the lines joining corresponding points all meet in a point, the centre of projection; and the point in which the two bases of the rows intersect as a point in the first row coincides with its corresponding point in the second. This follows from the definition. The converse also holds, viz.:— 'l‘iii-:oi:F..i.—If two projective rows have such a position that one point in the one coincides with its corrcspomlz'ng point in the other, then. tlzcy are prrspi-ctz'i~e, that is, the lines join my corrcsponding points all. pass through a common point, and form aflat pencil. Proof.—Let A, B, C, l) . . . he points in the one, and A’, B’, C’, l)’ . . . the corresponding points in the other row, and let A be made to coincide with its corresponding point A’. Let S be the point where the lines BB’ and CC’ meet, and let us join S to the point D in the first row. This line will cut the second row in a point D”, so that _-,ll,C,D are projected from S into the points A,B’,()’, D”. The cross-ratio (ABCD) is therefore equal to (AB’(,"D”). and by hypo- thesis it is equal to (A’B’C’D’). Hence (A’B’C’D”)=(A’B’C’D’), that is, D” is the same point as D’. § 31. Theorcm.—If two projected flat pencils in the same plane are perspective, then the intersections of corresponding lines form a row, and the line joining the two centres as a line in the iii-st pencil corresponds to the same line as a line in the second. And con vcrscl _v, TIIEOHI-1.I.--—If two projective pencils in the same plane, but with Il{,I7I:'r7zt centres, have one line in the one coz'ncz'dent 1I:ith its corre- sponding line in the other, then the two pencils are pcrspecti've, that is, the intersection of corresponding lines lie in (L Zinc. GEOMETRY 393 The proof is the same as in § 30. § 32. Theorem.—lf two projective flat pencils in the same point (pencil in space), but not in the same plane, are perspective, then the planes joining corresponding rays all pass through a line (they form an axial pencil), and the line common to the two pencils (in which their planes intersect) corresponds to itself. And con- vcrsely, Theorem.—If two flat pencils which have a common centre, but do not lie in a common plane, are placed so that one ray in the one coincides with its corresponding ray in the other, then they are perspective, that is, the planes joining corresponding lines all pass through a line. . § 33. Thcorem.—If two projective axial pencils are perspective, then the intersection of corresponding planes lie in a plane, and the plane common to the two pencils (in which the two axes lie) corresponds to itself. And conversely, Theo?-r1n.——If two projective axial pcneils are placed in such a position that a plane in the one coincides with its corresponding plane, then the two pencils are perspective, that is, corresponding planes meet in lines which lie in a plane. The proof again is the same as in § 30. § 34. These theorems relating to perspective position become illusory if the projective rows of pencils have a common base. We then have :— T/irorcm.—In two projective rows on the same line—and also in two projective and concentric flat pencils in the same plane, or in two projective axial pencils with a common axis—every element in the one coincides with its corresponding element in the other as soon as three elements in the one coincide with their corresponding elements in the other. Proof (in case of two rows). —Between four elements A, B, C, D and their corresponding elements A’, B’, C’, D’ exists the relation (ABCD) =(A’B’C’D’). If now A’, B’, C’ coincide respectively with A, B, C, we get (ABCD)=(ABCD’) ; hence D and D’ coincide. The last theorem may also be stated thus :— Thcov-e2n.——In two projective rows or pencils, which have a com- mon base but are not identical, not more than two elements in the one can coincide with their corresponding elements in the other. Thus two projective rows on the same line cannot have more than two pairs of coincident points unless every point coincides with its corresponding point. It is easy to construct two projective rows on the same line, which have two pairs of corresponding points coincident. Let the points A, B, C as points belonging to the one row correspond to A, B, and C’ as points in the second. Then A and B eo- inci(le with their cor- p)espop(l1iiig ppiiitis, u t oes no . is, however, not necessary that two such rows have twice a point coincident with its corres wond- ing point; it isl‘ pos- sible that this hap- pens only once or not at all. Of this we shall see plenty of examples as we go on. § 35. If two projec- tive rows or pencils Fig. 8. are in perspective position, we know at once which element in one corresponds to any given element in the other. If p and q (fi0‘. 8) are two projective rows, so that K corresponds to itself, and if we know -5‘ that to A and B in )1 correspond A’ and B’ in (1, then the point S, where AA’ meets BB’, is the centre of projec- tion, aiid hence, in order to find the point C’ corresponding to C, we have only to join C to S; the point C’, where this line cuts (1, is the point required. If two flat pencils, S, and S2, in a plane are perspective (fig. 9), we need only to know two Fig. 9. pairs, at, a’ and b, b’, of corresponding rays in order to timl the axis . . . . I - , , , ' s of projection. This being known, a ray 0 1I1_ S-3. C011‘9S1l0-ldmg
K. —— 50