210 Mr Richmond, On the condition that five straight lines 



On the condition that five straight lines situated in a space of 

 four dimensions should lie on a quadric. By H. W. Richmond, 

 M.A., Fellow of King's College. 



[Received 22 November, 1899.] 



The customary symbol 8 n being used to denote a space of n 

 dimensions, let a, b, c, d, e, stand for five straight lines situated 

 in $ 4 . Each two of the lines lie in and determine an 8 3 , and 

 each three of the lines are intersected by one line and one only, 

 viz. the line common to the three S 3 's determined by each two of 

 them : thus the lines a and b determine an S 3 , — denoted hereafter 

 by (ab), — which contains them ; and the three lines a, b, c are met 

 by a unique line (abc), the intersection of the three S 3 's (be), (ca), 

 (ab). 



A quadric in $ 4 , (that is to say a locus in $ 4 whose points 

 satisfy a single condition of the second order), may be made to 

 satisfy fourteen conditions ; one quadric for example will pass 

 through fourteen arbitrarily chosen points ; and again, since to 

 contain a straight line in its entirety imposes three conditions 

 on a quadric, one quadric will pass through four arbitrarily 

 chosen straight lines and two arbitrarily chosen points. Clearly 

 then the five straight lines a, b, c, d, e, do not as a rule lie on 

 a quadric, and, in order that they may do so, a condition has 

 to be fulfilled. It is a geometrical form in which this condition 

 may be expressed that I propose to investigate. 



Supposing that a, b, c, d, e the five lines in S 4 lie on a quadric 

 q, I imagine the $ 4 immersed in and surrounded by a space of five 

 dimensions, 8 5 . Through q there pass an infinity of quadrics in 

 the S 5 , any one of which I select and denote by Q. Now a 

 quadric in $ 5 has properties recalling those of quadrics in space 

 of three dimensions : there lie on it an infinite number of $ 2 's, 

 which form two distinct families : two S 2 's, one from each family, 

 pass through every line that lies on the quadric. Two $ 2 's lying 



