1894.] of 27 Straight Lines upon a Cubic Surface. 247 



At a considerable distance from the tetrahedron : 



The straight lines in the first sheet in order are 2, 5, 14, 25, 6, 

 12, 21, 24, 15, 10, 20, 3. 



In the second sheet we find, 4, 22, 27, 18, 17, 13, 23, 1, 19, 9, 

 7, 8, 26, 11, 16. 



Art. 3. The general equation given by Ta3']or may be put 

 into the form 



(X + I) {M+ m) {N+ n) (P -i p) xyzu = {-lx + My ■\-Nz + Pu) x 

 {Lx - my -\-Nz + Pu) (Lx + My -nz + Pu) (Lx + My + Nz -pu). 



It is evident that this surface can be constructed if we 

 determine the ratios the constants L, M, N, P, I, m, n, p bear 

 to one another. 



If the tetrahedron of reference be a regular one, having all the 

 edges equal, certain points of intersection seven in number taken 

 upon the edges of the tetrahedron divide them in the ratios of 

 these constants. If the tetrahedron be not regular these points 

 still determine the ratios of the constants, but the lengths of the 

 edges have to be considered. 



Take the tetrahedron as regular, the base being ADC, B the 

 centre of the triangle marking the projection of the vertex of the 

 tetrahedron upon the plane of ADC. 



In BD mark the point (12, 10). This determines the ratio M : p. 



In BD produced (2, 8) M : P. 



In ^D produced (4, 9) L : P. 



n .P. 



N:P. 



l:N. 



L : m. 



InDG (8,9) 



In i)(7 produced (6, 1) 



In AC (1, 3) 



InBA (4,5) 



(4, 9), (4, 5) pi'oduced gives (4, 6) in BD, 



(6,1), (1,3) (1, 2)in^i>, 



(8, 2), (2,1) (2, 3) in ^5, 



(4, 9), (8, 9) (7, 9) in ^6', 



(8, 2), (8, 9) (7, 8)in5C, 



(7, 9), (7, 8) (7, 12) in AB, 



(7, 12), (10, 12) (11, 12) in AD, 



(6, 1), (4, 6) (o, Q) in BC, 



(5, 6), (4, 5) (5, 11) in ^C, 



(5, 11), (11, 12) (10, 11) in DC, 



(1, 3), (2, 3) (3, 10) in BC. 



