PROFESSOR CAYLEY ON CUBIC SURFACES. 
293 
this is 
4 a 3 y\y-x)(y-z)(y-w) 
+a 2 {(12y-^-(8/3 7 +36% 3 -f(30i3H8y% 2 -36y^+27§ 2 } 
+ 2a{(6y 2 -/3V-9i3% 2 +(12/3^-2/3y 2 -18y%+2y 3 +27B 2 -9/3yS} 
—(x— zf(x — wf(z — wf — 0 . 
123. The nodal curve is made up of the lines (y=x=z), (y=x- =w), ( y—z—vo ), reci- 
procals of the three transversals. 
To show this I remark that, writing 
0 — {x—y) + {*-y)+{w—y), 
Y'=(v-y)(z-y)+(x-y)(w-y)+(z-y)(w-y), 
V =(x-y)(z-y)(w-y), 
the equation of the surface may be written 
4 ay{y-x)(y-z){y-w) 
+a 2 {y% 12p!l'- y ' a )+x . 18 7 ^+27B' 2 } 
+ 2 a{y(- 6/3' 2 ^ + 2/3'y' 2 + 9y'S')+2y' 3 -f 2 7S' 2 - 9)3 V S' } 
— ■ «) 2 (^ — w) 2 (2 — w) 2 = 0 , 
whence observing that y' is of the order 1 and o' of the order 2 in (x—y), (z—y) con- 
jointly, each term of the equation is at least of the second order in (x—y), (z — y) con- 
jointly; 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=w — y, and therefore x—y-\-p, z—y-\-r, 
w—y-\-s, the cubic function (putting therein T=V-j-;?/U) becomes 
«(Y+?/U)Y 2 +(y-pU)(y-rU)(y-sU) ; 
writing j3', y, l' =p-{-r-\-s , pr-\-ps-\-rs, prs, the coefficients are (3(«-fl), ay—[ 3', y', — 3&'), 
and the equation of the surface is thus obtained in the form 
27(a + l) 2 S' 2 
+i8(a+i)(ajr-/yyy 
+ 4(«+l)y' 3 
- Uay—^jy 
- (ay-p)Y=0, 
which, arranging in powers of a, and reversing the sign, is the foregoing transformed 
result. 
125. The cuspidal curve is given by the equations 
3(<z-fl), ■ 
-2 ay -(3, ay 2 f y 
1 
to 
y 
l 
3 a 
ay 2 +y, | 
