GENERAL EQUATIO^^ OF A CUBIC SGRFACE. 
49 
the equations 
— .T + (\ + ajh) ?/ = 0 
c/6 2 / + jaw = 0 
ax — 1/6) y + 8w = 0 
are simultaneously true. 
Hence 
— 1 \ + ail) . 
j , c/6 jji =0. 
a ^ — \lh 8 
Again, because this line intersects 10, whose equations are 
u — r/T = 01 
the equations 
\y + ajd . w = 0 
— 2 + (/a + c/d) U = 0 
/3y + yz 4- (S — 1/d) u = 0 
are satisfied simultaneously. 
Hence 
X . a/d. 
. — 1 fji “h cjd 
jS y 8 — l/d 
These equations of condition may be written as follows :— 
a6Xja dr — l)/^ — c8 = 0 
yd'KjL -|- (cy “h — l) X “ ci/3 = 0 
It is clear that the values of X and /x are the roots of the equations 
cycZ8X + (a6X + aa + 6/8 — 1) {(cy + d8 — l) X — aji] — 0, 
aah/Sjx -|- {ydfi + cy fi- dB — 1 ) {(c^a + 6/3 — l) /x — c8} = 0 , 
respectively. 
It is also clear that the roots of these equations must be so chosen that they 
satisfy the equation 
{ahXfjL — c8) {ydXfx — a/3) = [aa +6/8 — 1) (cy + (r/8 — 1) 
MDCCCXCIV.—A. H 
