parallel to the axis will be the lines i' and j. At the same time a parallelogram S]'I'I'S has been formed. If now the plane rr' be turned about the axis, then the points I' and I will not move in their planes; hence the lengths Tj and TI', and therefore also SI and SJ, will not change. If the plane rr is kept fixed in space the point will remain fixed, and S describes a circle about ] as centre ant; with S] as radius. This proves the last part of the theorem in 5.

§ 8. The plane -/r' may be turned either in the sense indicated by the arrow at Z or in the opposite sense till -rr' falls into -rr. In the first case we get a figure like fig. 3; i' andj will be on the same side of the axis, and on this side will also lie the centreS; and T ij 1,

J' J s

s 1

s" I """

S J J

FIG.;3. Flo. 4.

then ST=S]-4-SI' or SI'=]T, S] =I'T. ' In the second case (fig. 4) i' and j will be on opposite sides of the axis, and the centre S will lie between them in such a position that I'S=T] and I'T =S]. If I'S=S], the point S will lie on the axis. It follows that any one of the four points S, T, J, I' 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 determine the axis.

§ 9. We shall now suppose that the two projective planes rr, rr are perspective and have been made to coincide. If the centre, the axis, and either one pair of corresponding points on a line through the centre or one pair of corresponding lines meeting on the axis are given, then the whole projection is determined. Proof.-If A and A' (fig. I) are given corresponding points, it has to be shown that we can find to every other point the corresponding poirit B'. join AB to cut the axis in R. ]oin 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 the theorem of § 4. For, if for any point C the corresponding point C' be found, then the triangles ABC and A'B'C' are, by construction, co-linear, hence co-axal; and 5 will be the axis, because AB and AC meet their corresponding lines 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.

§ Io. Rows and pencils which are projective or perspective have been considered in the article GEOMETRY (G. §§ I2-40). All that has been said there holds, of course, here for any pair of corresponding 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 independent of the perspective position:-

The correspondence between two projective rows or pencils is completely 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 corresponding points, then their cross-ratios are equal (AB, CD)=(A'B', C'D')-where (AB, CD) =AC/CB:AD/DB.

If in particular the point D be at infinity we have (AB, CD)= -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 infmity of the axis unless the axis is altogether at infinity. H€HCf* In two perspective planes every row which 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 the axis is at infinity then every row is similar to its corresponding row.

In either of these two cases the metrical properties are particularly 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). l

In both cases the mid-points of corresponding segments will be corresponding points.

§ II. Involution.-If the planes of two projective figures coincide, then every point in their common plane has to be counted twice, once as a point A in the figure rr, once as a point B' in the figure 1r'. 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 seq.). If two projective planes coincide, and if 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 denoted by A in the first plane and by 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 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 oint 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 point where XB cuts SD'.

But this is the same point as C'.

This point C' might also be

got by drawing CB and joining

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 corresponding points A and A' have to be harmonic conjugates with regard to S and the point T where AA' cuts the axis.

I f two perspective figures be in involution, two corresponding points are harmonic 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 be 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 involution. § 12. Projective Planes which are not in perspective position.-We return to the case that two planes rr and rr' are projective 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. 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 that 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. 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 correspondence 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 projective in the old sense (G. §§ 25, 30). Further- f two planes are projective to a third they are projective to each other. The correspondence between two projective planes rr and -/r' is determined we have given either two rows u, v in rr and the corresponding rows u, v' in 1r', the point where u and v meet corresponding to the points where u' and v' meet, or two pencils U, V in rr and the corresponding pencils U', V' in 1r', the ray UV joining the centres of the pencils in rr corresponding to the ray U'V'.-It is sufficient to prove the first part. Let any line a cut u, 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 corresponding point.

§ 13. If the planes of two projective figures coincide, and if either four points, of which no three lie in a line, or else four 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 jigures 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 S

A CD1

s X s

T

CID

O

A B

~ FIG. 5.

its intersection Y with the axis