GENERAL EQUATION OE A CUBIC .SURFACE. 
51 
These equations of condition may be written 
ah\v + (aa hfi — l) y — yd ~ 0 
ch\v {cy dS — l) \ — (1,8 = 0 
respectiveJy. 
Hence the values of X and i/ are the roots of the equations 
cydSX -f- {cd)\ “h ~t“ ” 1) {{py “h ^ — b 
aoib^v + {chv + cy + c/S — 1) {{cw + ?>y8 — 1 ) 1 ^ — yd] — 0 
respectively. 
It will be observed that the equation to find X in determining- the equations of 18 
and 19 is identical with the equation to find X in determining the equations of 16 
and 17. It appears, therefore, that one of the two lines 18 and 19 lies in the plane 
of 1 and 16, and the other in the plane of 1 and 17. Here we assume that the 
complanar sets are 1, 16, 19, and 1, 17, 18. 
In an exactly similar manner we can prove that each line of one pair intersects one 
or other of the lines of the second pair in the case of each of the sets of pairs— 
1., in.; 1., IV. ; 1., vi. ; in, 111.; in, v. ; in, vn ; iin, iv ; iin, v. ; 
iv., V. ; iv,, vi. ; and v., vi. 
§ 8. There are three other sets of pairs, to which a different method of proof must he 
applied, viz., h, v. ; in, iv. and iii., vi. Let us consider tlie lines of the pair v., that is, 
the lines 24 and 25, which intersect the lines 3, 6, 9 and 12. 
Any line intersecting 3 and 9 is represented by equations of the form 
X — ((T = (f)u 
z — cT = 
(24) or (25). 
Since this line intersects (6), whose equations are 
y-6T = 0l^ 
.1- = oj ’ 
the equations 
iijb . j -fi cf)U — 0 
{xjj ctb) 1/ — 2: = U 
(/3 — 1/b) u yz d- = 0 
are simultaneously true. 
