314 PEOFESSOE C. J. JOEY ON QUATEENIONS AND PEOJECTIVE GEOMETEY. 
146. It may be of interest to show how we can fully account for the lines on the 
cubic surface (535). Let the six points in which the critical curve (r) = 0 cuts 
the plane S/(/ = 0 be denoted by the symbols 1, 2, 3, 4, .5, 6 ; and let (12), (23), &c., 
denote the fifteen connectors of these points. Further let [ 1 ], [ 2 ], ... [ 6 ] denote 
the six conics that can he drawn through all hut one of the six points. 
The curves and points represented hy these 27 symbols transform into the lines on 
the cubic. By (531) and (533) we account for the lines and the points. In general 
a unicursal curve transforms into a curve of thrice the order, but for every inter¬ 
section with the critical curve a line breaks off. Thus the six conics likewise 
transform into lines. 
Any jDair of these loci, which intersect in a point which is not critical, continue to 
intersect after transformation, and this consideration enables us to write dovm the 
full scheme of double-sixes on the cubic surface. These fall into three types :— 
/I 2 3 4 5 6 
• l[l] [2] [3] [4] [5] [6] 
,. / 1 2 3 (5b) (64) (•15)\ 
• 1(23) (31) (12) [4] [5] [ 6 ];- 
ttt /I [1] (23) (24) (25) (26)\ 
• (2 [2] (13) (14) (15) (16)/ 
In these schemes, every line represented by a symbol in one row intersects eveiy 
line in the other row, except that denoted by the symbol in the same column. There 
are thus 36 doulde-sixes ; one of the first type, twenty of the second, fifteen of 
the third. 
The schemes are easily oljtained ])y taking two ntm-intersecting lines, say 1 and [l], 
when we have 
1 intersects (12), (13), (14), (15), (16), [2], [3], [4], [5], [ 6 ], 
[J] „ (12), (13), (14), (15), (16), 2, 3, 4, 5, 6, 
and, discarding the common lines, the double-six is found. In like manner the 
45 triple tangent planes belong to one or other of the types 
(1,[2], (12)) or ((12), (34), (56)). 
147. One or two relations respecting a point on a critical curve and its line 
homograph may be mentioned. Since the line (531) has a point for its homogi'aph, it 
must 1)6 a tri|Ie cliord of the sextic fi’, (r) = 0 . It meets this sextic in three points, 
'P\, p 2 i Psj intersects the Jacobian in a fourth point pQ or (p). To the three 
points p^, p 2 , P 3 correspond the three triple chords of the q sextic which pass 
through q; and the homograph of every plane through the line pi, jo.i. pg is a cubic 
