421 
412] 
A MEMOIR ON CUBIC SURFACES. 
or substituting for /3, y, 8 in the first and last lines their values 
this is 
(= x + z + w, xz + xw + zw, xziu), 
4«Y (y -®)(y- z) (y - w) 
+ a 2 {(12 7 - /3 2 ) y 4 - (8/3 7 + 368) f + (30/38 + 8 7 2 ) if - 36y8y + 278-} 
+ 2a {(6 7 2 - /3 2 7 - 9/38) f + (12/3 2 8 - 2/3 7 2 - 18 7 8) y + 2 7 3 + 278 2 - 9/3 7 8} 
— (x — zf (x — w) 2 (z — w)- = 0. 
123. The nodal curve is made up of the lines (y = x = z), (y = x = w), (y = z = w), 
reciprocals of the three transversals. 
To show this I remark that, writing 
/3' = (x - y) + (z - y) + (w - y), 
y = (as- y) (z-y) + (as- y) (w -y) + (z-y)(w - y), 
% =(v-y)(z-y)(w-y), 
the equation of the surface may be written 
4ci 3 f (y-x)(y-z) (y-w) 
+ a 2 [f (12/3'8' - 7 2 ) + x. 18 7 8' + 278' 2 } 
+ 2a {;y (- 6/3' 2 8' + 2/3V 2 + 9 7 '8') + 2 7 ' 3 + 278' 2 - 9/3' 7 '8'} 
— (x — z) 2 (« — w) 2 — ic) 2 = 0, 
whence observing that y is of the order 1 and S' of the order 2 in (x — y), (z — y) 
conjointly, each term of the equation is at least of the second order in (x — y), (z — y) 
conjointly; or we have y = x = z, a nodal line; and similarly the other two lines are 
nodal lines. 
124. The foregoing transformed equation is most readily obtained by reverting to 
the cubic in T, U, viz. writing p = x — y, r = z — y, s — io — y, and therefore x = y + p, 
z = y + r, w = y + s, the cubic function (putting therein T= V + yU) becomes 
a(V+yU) V* + (V-pU)(V-rU)(V-sU)-, 
writing /3', y, 8' = p + r + s, pr + ps + rs, prs, the coefficients are (3 (a + 1), ay - /3', y, — 38'), 
and the equation of the surface is thus obtained in the form 
27 (a + l) 2 8' 2 
+ 18 (a + 1) (ay - ¡3') y'8' 
+ 4(a + l) 7 ' 3 
— 4 (ay — (3'f 8' 
- (ay - ¡3J y' 2 = 0, 
which, arranging in powers of a, and reversing the sign, is the foregoing transformed 
result.