316 
[459 
459. 
ON THE DOUBLE-SIXERS OF A CUBIC SURFACE. 
[From the Quarterly Journal of Pure and Applied Mathematics, vol. x. (1870), 
pp. 58—71.] 
The 27 lines on a cubic surface include, and that in 36 different ways, a double 
sixer; viz. a system of two sets of six lines 1, 2, 3, 4, 5, 6 ; V, 2', 3', 4', 5', 6', such 
that every line of the one set intersects all the non-corresponding lines of the other 
set, thus 
1 2 3 4 5 6 
1' 
2' 
3' 
4' 
o' 
6' 
there being in all 30 intersections. 
Any line say 4, of the one set, intersects five lines 1', 2', 3', 5', 6' of the other- 
set; and these six lines being given the double-sixer may be constructed; viz. (besides 
the line 4) we have a line 1 meeting the lines 2', 3', 5', 6'; a line 2 meeting the 
lines 3', 5', 6', 1'; a line 3 meeting the lines o', 6', 1', 2'; a line 5 meeting the lines 
6', T, 2', 3'; and a line 6' meeting the lines V, 2', 3', o'; and then the lines 1, 2, 3, 5, 6 
are all of them met by a single line 4', which completes the system. 
We may, if we please, consider the lines 4, 2 as given, and then 1', 3', o', 6' will 
be any four lines each of them meeting the two given lines 4, 2; 2' will be any 
line meeting 4; and we have to determine a line 4' meeting 2, such that there may 
exist the lines 1, 3, 5, 6, completing the system as above. Or what is the same 
thing, we have a skew quadrilateral 1', 2, 3', 4; 5' and 6' meet 2 and 4; 2' meets