GENERAL EQUATION OF A CUBIC SURFACE. 
59 
26, 27, 5, 2, and 9, and the line 3 cuts the lines 26, 27, 5, 2, and 10 ; whereas both 
the lines 3 and 4 cut the lines 26, 27, 5, 2, and 13. 
Such a set of lines as 26, 27, 5, 2, 9, is called a ‘‘pentad.” 
Again, from figure C, we see that there is but one line 10, which does not cut 
the pentad 26, 27, 5, 2, 9. 
Such a set of lines as 26, 27, 5, 2, 9, 10, is called a “ hexad.” 
We may summarize the last results by saying that the number of the lines of the 
surface which do not cut— 
a single 
line on 
the surface. 
. is 
16; 
either line of a 
non-intersectino’ duad 
♦ 99 
10 ; 
any 
99 
,, triad 
* 99 
6 ; 
9 9 
,, tetrad . 
* 99 
3 ; 
99 
,, pentad . 
* 99 
1 ; 
9 9 
99 
,, hexad . 
• 99 
0. 
Similarly, by inspection of the top six 
rows of Figure C, we conclude that— 
10 lines on the surface cut a definite line on the surface, and 16 do not. 
5 lines cut both the lines of a duad ; 10 lines cut 1 ; and 10 cut neither. 
3 lines cut all the lines of a triad ; 6 lines cut 2 ; 9 lines cut 1 ; and 6 cut 
none. 
2 lines cut all the lines of a tetrad ; 4 lines cut 3 ; 6 lines cut 2 ; 8 lines 
cut 1 ; and 3 cut none. 
1 line cuts all the lines of a pentad ; 5 lines cut 4 ; 10 cut 2 ; 5 cut 1 ; and 
1 cuts none. 
No lines cut all the lines of a hexad; 6 lines cut 5 ; 15 cut 2; none cut 
1 only ; and none cut none. 
We are now enabled to find the number of different duads, triads, &c. 
"NTiimLAi’ rlnorla 
27 . 
16 
= 216; 
1 . 
2 
triads 
27 . 
16 . 
10 
= 720; 
9y 
1 . 
2 . 
3 
tetrads . 
27 . 
16 . 
10 . 
6 
= 1080; 
99 
1 . 
2 . 
3 
4 
pentads . 
27 . 
16 . 
10 . 
6 . 
o 
= 432 ; 
99 
1 . 
2 . 
3 . 
4 . 
, 5 
hexads . 
27 . 
16 . 
10 . 
6 . 
2 . 
^ — 72. 
2 
4.5.6 
1 
