542 



1. It is a very well known property of the conic sections, 

 that if three of the four sides of a plane quadrilateral inscribed 

 in a given plane conic be cut by a rectilinear transversal in 

 three given points, the fourth side of the same variable qua- 

 drilateral is cut by the same fixed right line in a fourth point 

 likewise fixed. And whether we refer to the relation of invo- 

 lution discovered by Desargues, or employ other principles, 

 it is easy to extend this property to surfaces of the second 

 order, so far as the inscription in them oi plane quadrilaterals 

 is concerned. If then we merely wish to pass from one point 

 p to another point R of such a surface, under the condition 

 that some other point Q of the same surface shall exist, such 

 that the two successive and rectilinear chords, pq and qr, 

 shall pass respectively through some two given guide-points^ 

 A and B, internal or external to the surface ; we are allowed 

 to substitute, for this pair of guide-points, another pair, 

 such as b' and a', situated on the same straight line ab ; and 

 may choose one of these two new points anywhere upon that 

 line, provided that the otherhe then suitably chosen. In fact, 

 if c and c' be the two (real or imaginary) points in which the 

 surface is crossed by the given transversal ab, we have only 

 to take care that the three pairs of points aa', bb', cc', shall 

 be in involution. And it is important to observe, that in 

 order to determine one of the new guide-points, B'or a', when 

 the other is given, it is by no means necessary to employ the 

 points c, c', of intersection of the transversal with the surface, 

 which may be as often imaginary as real. We have only to 

 assume at pleasure a point p upon the given surface ; to draw 

 from it the chords paq, qbr; and then if a' be given, and b' 

 sought, to draw the two new chords ra's, sb'p ; or else if a' is 

 to be found from b', to draw the chords pb's, sa'r. For ex- 

 ample, if we choose to throw off" the new guide-point b' to in- 

 finity, or to make it a. guide-star, in the direction of the given 

 line ab, we have only to draw, from the assumed initial and 

 superficial point p. a rectilinear chord PS of the surface, which 



