of lines by projection from four dimensions 137 



planes through an arbitrary point 0, to meet a, b, e, d, meet the 

 line e in T and U, then the two planes which can similarly be 

 drawn through to meet the lines a', b', c', d', meet the line e 

 in the same two points T and U. In general two planes in four 

 dimensions have only one point in common; when they have two 

 points in common, the join of these points lies in both the planes 

 which then both lie in the same threefold space. By what we have 

 said there is a plane through OT intersecting a, b, c, d and also 

 a plane through OT intersecting a', b', c', d', with a similar state- 

 ment for planes through OU. Namely considering the two planes 

 through which meet a, b, c, d and also the two planes through 

 which meet a', 6', c', d' either one of the former meets one of the 

 latter in a line. 



To prove these statements we may proceed as follows. The 

 joining line of two points arbitrarily taken respectively, say, on the 

 lines b and c, will meet the space {a, d) in a point, from which, in 

 this space, a transversal can be drawn to a and d. Then the plane 

 of the original join and this transversal is a plane, say w, meeting 

 the four lines a, b, c, d. The point of intersection of these two lines 

 determining this plane m is evidently on the plane, a, common to 

 the spaces (6, c) and {a, d). Similarly the point of intersection of 

 the plane m with the plane, ^, common to the spaces (c, a) and 

 (6, d), is a point from which two transversals can be drawn respec- 

 tively to the pairs of lines c, a and b, d; and the plane of these 

 transversals is a plane through this point meeting the four lines 

 a, b, c, d; conversely the join of the two points where the plane C7 

 meets the lines c and a lies in the space (c, a), and so intersects the 

 plane /S, namely in the supposed unique point common to w and j8; 

 this join is thus identical with the transversal drawn from the 

 point (m, ^) to the lines c, a. There is thus an unique plane m, 

 meeting a, b, c, d, passing through any general point of the plane «, 

 beside the plane a itself. It will follow from the general result 

 enunciated above, to be proved below, that the plane w', drawn 

 through the same point of the plane a to meet the lines a', b' , c' , d' , 

 meets C7 in a line intersecting the line e. 



Take now any general point 0, and a varying point P of the 

 line d; a plane can be drawn through OP to meet the lines a, b, 

 this being the plane containing OP and the common transversal 

 of OP, a and b. Let this plane meet a, b respectively in P^ and P^. 

 Thereby any position of P, on the line d, determines the position 

 of Pi on the line a. Conversely given and Pj, a plane can be 

 drawn through OP^ to meet 6 and d, which, being unique, coincides 

 with the former. Thus any position of Pj on a determines the 

 position of P on d. The correspondence being algebraic, it follows 

 that Pj, P describe homographic ranges respectively on a and d. 

 Using the line c instead of b, we obtain another range (P') on d. 



