242 Mr Blythe, On the Construction of a model [Oct. 29, 



The necessary drawing is however very complicated and diffi- 

 cult, but if we take an equation given by Cayley, and suppose the 

 tetrahedron of reference to have three edges equal and at right 

 angles to each other, and further take another plane (one of the 

 forty-five) at right angles to the base of the tetrahedron, the 

 construction is much simpler. 



Table of Reference. 



[Numbers indicate straight lines. Three numbers in a bracket 

 indicate three straight lines in a. plane.] 



(4,6,5), (13,10,3), (9,8,7), (4,13,9), (6,10,8), (5,3,7), 

 (12, 25, 18), (24, 14, 17), (19, 16, 1), (12, 24, 19), (25, 14, 16), 

 (18,17,1), (2,21,22), (20,15,27), (23,26,11), (2,20,23), 

 (21,15,26), (22,27,11), (4,12,2), (18,14,15), (9,1,11), 

 (5, 14, 11), (3, 1, 2), (7, 12, 15), (6, 15, 1), (10, 11, 12), (8, 2, 14), 

 (4, 27, 16), (13, 23, 18), (9, 21, 24), (4, 26, 17), (13, 22, 19), 

 (9, 20, 25), (5, 18, 21), (3, 24, 27), (7, 16, 23), (6, 22, 25), 

 (10, 20, 17), (8, 26, 19), (6, 23, 24), (8, 27, 18), (10, 21, 16), 

 (5, 19, 20), (3, 25, 26), (7, 17, 22). 



N.B. To find whether two straight lines intersect, observe 

 whether they occur in the same plane. 



One side of every triangle intei'sects with one side of any other 

 triangle. 



Art. 1. Geometrical construction to find the twenty-seven 

 straight lines. 



Art. 2. Notes on the form of the surface. * 



Art. 3. Geometrical construction to find the twenty-seven 

 straight lines from the general equation given by Taylor. 



[Fig. I. re^iresents the plane of the base of the tetr^ahedr-on of 

 reference, the straight lines 4, 9, 13 being the edges of the base. 

 On the sides of this diagram are marked the points at luhich the 

 projections of the straight lines cut them. 



The rings (7, 9), (8, 9) indicate where the' straight lines 7 and 8 

 cut 9. (18, 1); (11, 12); (9, 25) are the projections of these points 

 upon the plane of the figure. 



Fig. II. represents a plane containing the straight lines 9, 20, 25 

 indicated by numerals placed at the end of them, and is at right 

 angles to I. 



Figs. III. and IV. are two planes representing faces of the 

 tetrahedron of reference, also at right angles to I. containing respec- 

 tively the straight lines 4, 12, 2 and 13, 14, 15. 



In Figs. II., III., IV. the numbered points indicate where the 

 straight lines meet these sections.] 



