Wl 



Henderson— A Memoir. 



125 



, (12), (13), (14), (15) = 



(21), , (23), (24), (25) 



(31), (32), , (34), (35) 



(41), (42), (43), , (45) 



(51), (52), (53), (54), 



this being- the condition* that any five lines 1, 2, 3, 4, 5 say, 

 have a common tractor, where the equations of lines i and j 

 are 



at x + h y + ci z + di w — ) 

 <*i * -f- ft y -f yi z -f- S t w = \ 



(O 



and 



«i # -f- fy .y + 3 * + ^ w = ) 



= f 



cy * + ft J + *,- Z + % W = f • ' ' O ) 



respectively, and we understand by ( z)' j the determinant 



«i j #i j Ci , <^- 

 a* , ft i Yt i 8 i 



«* 



fy 5 



The fine lines 1', 3', 4', 5', 6' constituting- the co-tractorial 

 quintuple do not mutually intersect, since in forming the 

 determinants ( ij ), none of them are found to vanish. A pos- 

 sible difficulty arises from the fact that the hyperboloid through 

 any three of these five collinear lines mig-ht touch a fourth, 

 that is to say that certain four of the lines might have a dou- 

 ble tractor.t That such is not the case appears in the 

 sequel. 



Determining- now the common tractor, 2, of these five lines, 

 we find it to have the equations 



♦Sylvester, "Note sur l'involution de six lignes dans l'espace", Oomptes 

 Rer.dus, vol. LII (1861), pp. 815-817. 



fOayley, "On the Six Oo-ordinates of a Line". Trans. Oamb. Phil. Soc., 

 yol. XI, part II (1869) pp. 290-323. 



