454] 
A THIRD MEMOIR ON QUARTIC SURFACES. 
281 
X =7. (y'y" y - "z\ 
F = /8 W z-ij'x), 
Z =y (p'p"x-a'a"y\ 
X' = «' (y'y y-p"pz), 
Y' = /3' (a”a z-y"y x), 
Z' = y (/3"/3 x — a’a y), 
X" = a" (y y y - P ¡3' z), 
Y” = P" (a. cl z — y y' &•), 
^ =7" (Pp'x-CLa’y), 
A = x 2 + y 2 + — 2^ — 2a;?/, 
5 = aa'a" (t/ 2 ^ — yz 2 ) + /3/373" (z 2 x — zx 2 ) + yy'y" (&' 2 ?/ - xy 2 ) + Mxyz, 
C = (aa'a"yz + PP'P"zx + yy'y'xy ) 2 ; 
where 
M=(P — y ) a'a" + (y — a ) /3'/3" 4- (a — P ) y'y", 
= (/S' - y' ) a"a + (y' - a') /3"/3 + (a' - /3') y'y, 
= (P" — y") aa' + (y"-a")/3/3' + (a" -/3") y y', 
= -i {08-7)(S' -7)08" -7") + (7■-«)(7 - *') (7"-«") + (a -/3) («'- /3') (a"-/3")} ; 
also 
iT = 4nxa'a."PP'P"yyy" : 
and we have identically 
a; y z\ 
a" + W' + 7)‘ 
The 16-nodal Surface 16(1, 1, 1, 1, 1, 1). 
157. Rummer starts from an irrational equation, which is readily converted into 
the following 
fx (X — w) + y/y (Y — w) + y/ z (Z — w) = 0, 
and then, rationalizing, we have 
Aw 2 + 2 Bw + (7=0, 
where as above 
AC — B 2 = Kxyz PP’P". 
This agrees with the foregoing theory; viz., the point (x = 0, y = 0, z = 0) being a node, 
the rationalized equation must, of course, be in the node-form (1, 1, 1, 1, 1, 1), (being 
the only node-form); and the symmetry of the formulae enables us at once to write 
c. vii. 36