132 Journal of the Mitchell Society. [Nov. 



and have by our last theorem 



which may be phrased as follows: — 



The anharmonic ratio of the four points, on one of five 

 co-tractorial lines, -which are collinear -with three of the remain- 

 ing lines is equal to the anharmonic ratio of the four planes 

 determined by these remaining lines and their common tractor* 



For other interesting- results on the anharmonic ratios con- 

 nected with the double-six configuration consult the paper of 

 Kasner just referred to. 



§9. Five Co-tractorial Lines as Primitive. 



Given any five co-tractoriai lines, these determine uniquely, 

 as was shown in § 7, the double-six configuration. Then if 

 we consider the plane of ij' ', it will be met by the lines i, j in 

 points which lie on the line ij. Since 6 P a = 15, the 12 lines 

 of the double-six together with the 15 new lines make up 27 

 in all, the total number upon the cubic surface. t Then the 

 condition A $ = (§7,), which is the condition that five lines 

 be co-tractorial, is likewise the condition that five given lines 

 may lie in a cubic surface. The result of Sylvester, viz. that 

 A $ = is the condition that five given lines be co-tractorial, 

 is found in a paper on the "Involution of Six Lines", + a sub- 

 ject first studied by him in connection with a theorem in the 

 Lehrbuch der Statik, by Mobius (Leipzig). 



If we are given five lines, defined by their six co-ordinates 



*Kasner, Am. Journal Math. vol. XXV, No. 2 (1903), p. 114. 



t Sylvester, Oomptes Rondus, vol. LII (1861), pp. 977-980. Cf. also Sal- 

 mon, Geom. of Three Dimensions, 4th edition, pp. 600-501 and R. Sturm, 

 Flachen Dritter Ordnung, pp. 57-59. 



jOomptes Rendus, vol. LII (1861), pp. 815-817. 



