332 Mr. A. Cayley on the Homographic D-ansformation 



and V = intersects the surface x 2 -f y 1 4- z~ + w* — in four lines, 

 the surface V ; = will also intersect this surface in the same 

 four lines. And it is, moreover, clear that the line joining the 

 points (a?„ y v z v w,) and (# 2 , y 2 , z^ w^) touches the surface 

 V = in the point (£', rj', £', «'). We thus arrive at the theorem, 

 that when two points of a surface of the second order are so 

 connected that the coordinates of the one point are linear func- 

 tions of the coordinates of the other point, and the transforma- 

 tion is a proper one, the line joining the two points touches two 

 surfaces of the second order, each of thern intersecting the given 

 surface of the second order in the same four lines. Any two 

 points so connected may be said to be corresponding points, or 

 simply a pair. Suppose the four lines and also a single pair is 

 given, it is not for the determination of the other pairs necessary 

 to resort to the two auxiliary surfaces of the second order ; it is 

 only necessary to consider each point of the surface as deter- 

 mined by the two generating lines which pass through it ; then 

 considering first one point of the given pair, and the point the 

 corresponding point to which has to be determined, take through 

 each of these points a generating line, and take also two gene- 

 rating lines out of the given system of four lines, the four gene- 

 rating lines in question being all of them of the same series, 

 these four generating lines intersecting either of the other two 

 generating lines of the given system of four lines in four points. 

 Imagine the same thing done with the other point of the given 

 point and the required point, we should have another system of 

 four points (two of them of course identical with two of the points 

 of the first-mentioned system of four points) ; these two systems 

 must have their anharmonic ratios the same, a condition which 

 enables the determination of the generating line in question 

 through the required point : the other generating line through 

 the required point is of course determined in the same manner, 

 and thus the required point (i. e. the point corresponding to any 

 point of the surface taken at pleasure) is determined by means 

 of the two generating lines through such required point. 



It is of course to be understood that the points of each pair 

 belong to two distinct systems, and that the point belonging to 

 the one system is not to be confounded or interchanged with the 

 point belonging to the other system. Consider, now, a point of 

 the surface, and the line joining such point with its corresponding 

 point, but let the corresponding point itself be altogether dropped 

 out of view. There are two directions in which we may pass 

 along the surface to a consecutive point, in such manner that 

 the line belonging to the point in question may be intersected 

 by the line belonging to the consecutive point. We have thus 

 upon the surface two series of curves, such that a curve of each 



