[339 
195 
339] ON SKEW SURFACES, OTHERWISE SCROLLS, 
which putting therein p + q = a, pq = /3, give 
and thence 
terms accord- 
x', y, z\ w ) 
:d so as that and therefore 
and consequently 
which is right. 
2a 
= 
/3 + 
1 
2 
a 2 - 
+ 10, 
2a' 
= 
/3- 
1 
2 
a 2 + 
i a 
-10, 
2 A 
= 
-2 A' = 
- 
1 
2 
a 2 + 
\ a 
- 6, 
2aa' 
= 
№- 
1 
3 
a 3 + 
\ a2 
_ 103 
6 
a + 26, 
2 AA' 
= 
- 
1 
3 
a 3 + 
i a ' 
_ 73 
6 
a + 14, 
tA 
+ 
2a = 
/3 
— 
2a -4- 
■ 4, 
2 A' 
+ 
2a r = 
/3 
- 
4, 
2 AA' 
- 
2aa' = ■ 
~\a!3 
+ 
5a — 
12, 
Weight = (¡3 — 2a + 4) (/3 — 4) + i a/3 — 5a + 12 
= /3 2 — f a/3 + 6a — 8 
= i(/3-2)(2/3-3a + 4), 
S (m 3 ) = |/3x weight 
= */3 08-2) (2/8-3a+ 4), 
2. 
-1), 
2J-3) 
(2p - 3j, 
Annex No. 3.—Investigation of G (m 4 ) in the case where the curve m is a pq complete 
intersection (referred to, Art. 42). 
Suppose, as before, that U = 0, V = 0 are the equations of the two surfaces of 
the orders p and q respectively; taking also (x, y, z, w) as the coordinates of a point 
on the curve, and substituting in the equations x + px, y + py, z + pz, w + pw in place 
of the coordinates, then if A = x'd x + y'd y + z'd z + wd w , we have as before 
(AU, A 2 U, . . APUlfl, pf- 1 = 0, 
(AF, A 2 F, . . A^F][1, P y~ x = 0, 
where the numerical coefficients 1, _L-^, &c. are to be understood as before. 
Suppose now that (x, y, z, w) are the coordinates of a point on the curve, through 
which point there passes a line through three other points, or line G (m 4 ); and that 
(x', y', z', w') are the current coordinates of a point on such line; the two equations 
in p must have three equal roots; or we must have a system equivalent to three 
equations, or say a plexus of three equations. The coordinates (x, y', z', w'), although 
four in number, are in iact eliminable from this plexus; or what is the same thing, 
combining with the plexus the equation 
ax' + ¡3y' + yz + 8w' = 0 
25—2