1894.] of 27 Straight Lines upon a Cubic Surface. 248 



Art. 1. The following rules derived from the equation of the 

 surface are sufficient to determine the straight lines 4, 9, 13, 20, 

 25, 12, 2, 14, 15, 1 and 11. 



1. The heights of the intersections, (14, 25), (2, 20), (12, 15) 

 above the plane of (4, 9, 13) are in the proportion I : m : 1. In 

 this case 3:2:1. 



2. The straight lines 20 and 25 each divide the edge 9 in the 

 proportion Imn : 1. In this case n = 2^. 



The edges 4 and 13 are divided in a similar manner by 2, and 

 12, and 14 and 15 respectively in the ratios mn : 1 and In : 1. 



3. The projections of the straight lines 1, and 11 in Fig. I. 

 divide the edge 9 externally in proportion 1 : Im and the line 4 

 internally in the ratio 1 : w. 



4. Now to find any other straight line we proceed as follows. 

 Take any plane containing the required straight line (e.g. 18) and 

 two other known straight lines (e.g. 12, 25). 



Next take any other two triangles already found 9, 11, 1 and 

 14, 15, 13. 



We find that 25 intersects with 9, 12 with 11, and 18 with 1. 

 Since these are the intersections of the sides of two independent 

 triangles they are in one straight line. Therefore joining (9, 25) 

 to (11, 12) in Fig. I. and producing the straight line we find the 

 point at which 18 meets 1. 



Similarly by joining (14, 25) and (12, 15) in Fig. IV. we find 

 the point at which 18 meets 13. Join 18, 1 to 18, 13, the pro- 

 jection of 18 in I. is determined. 



It is known that 18 meets 25 and 12 from the table of refer- 

 ence, therefore we mark on 25 and 12 in II. and III. points vertically 

 above those in which its projection meets 9 and 4 in I. 



Measuring horizontal distances from I. and verticals from II. 

 and III. we can construct the position of 18 in a vertical plane 

 drawn through it, as in V. 



The remaining straight lines can be found in the same way, 

 using those already determined to fix the position of others. On 

 the plate opposite the straight lines are all given. 



The simplest method of constructing a model of the straight 

 lines is as follows. 



First make drawings of I., II., III. and IV. to a scale of three 

 times that given in the plate, being very careful that the lines 

 cut each other exactly in the ratios stated. 



Next draw each of the remaining lines to the same scale in 

 the same manner as in V. Then take a drawing-board of any 

 convenient size, probably it may be better to make the model 

 exactly double the scale of the drawings for convenience of 

 measurement. 



