164 Mr J. Larmor, On Possible Systems of Jointed [May 26, 



A A 



a i«i • & A cos c l7l - a 2 v 2 . 6 3 /3 3 cos c 3 7 3 



= I «A & 2 + & 2 - V - /3 3 2 ) - \ &A « + « 2 2 - a* - O, 

 and we therefore have 



a. or.. 6.5, ax.&A 



c i7i c s 7, 



therefore rtl 6 A . a^ = a^ . a 2 ^ 7l . 



And in the same way we can obtain three other similar relations, 

 thus making up the four relations required. 



Having now obtained these relations between the segments of 

 two triads of mutually intersecting lines in space, we may easily 

 verify their truth in other ways. We notice that they are projec- 

 tive for the same reason that anharmonic ratios are projective. 

 Projecting therefore on the principal plane of the hyperboloid to 

 which they belong, we have two triads of tangent lines to a plane 

 ellipse. We can now project the ellipse into a parabola. But 

 three fixed tangents to a parabola cut all variable tangents simi- 

 larly, since they with the tangent line at infinity cut them in 

 a constant anharmonic ratio: hence now 



a t : b % : c x = a 2 : \ : c 2 = a 3 , : b 3 



and the relations are obviously true. 



[We may express this argument differently by changing the 

 hyperboloid into a hyperbolic paraboloid by a linear transforma- 

 tion (which we may call a projection in space of four dimensions), 

 and noticing that the theorems are true for the paraboloid because 

 the generators of one system 1 divide all those of the other system 

 similarly.] 



It is to be noticed that they are not true in general for two 

 triads in a plane : also, inasmuch as there is only one condition 

 necessary that six lines should touch a conic, that three other 

 relations do hold in a plane. 



When three lines cross three other lines in a plane the three 

 relations between the segments formed are however still true for 

 lines crossing in space, and are moreover clearly of a projective 

 character. We may obtain one of them as follows. From the 

 equations between the cosines already given, we find 



where 



a A' 

 cos a 2 a 2 = -^ 



rtjOt, — a 2 a 2 

 c x 7i ° - %% 



