124 
ON THE SURFACES THE LOCI OF 
[503 
where P', Q', R', S' denote h'y — g'z + a'w, &c., and where, finally, x, y, z, w are to be 
replaced by ux a + vx b , &c. Since for these values P, Q, R, S vanish, the expression 
becomes 
= - 2vu (A AP + BAQ + CAR + DAS) 
+ AP+BQ' +CR' +DS 
that is 
= A{P’- 2uvAP) + B(Q' — 2uvAQ) + C(R' — 2uvAR) + D (S'- 2uvAS) 
and we have, in fact, P' — 2uvAP = 0, &c. For, writing for a moment 
x, y, z, w = ux a + vx b , uy a + vy b , uz a + vz b , uw a + vw b , 
X, y', z\ w = ux a - vx b> uy a -vy b , uz a -vz b , uw a - vw b ; 
then, for instance, 
where 
and thence 
S' = &'x + h'y + cz, 
a', b', c' = Yz — Zy', Zx — Xz, Xy' — Yx ; 
S' = - 
X, 
Y, Z 
x, y , z 
x', y', z’ 
= 2 uv (aX + bF+ c Z) 
= 2uvAS; 
and similarly for the other equations. The factor is thus = 0 2 ; there is only one such 
factor, and the line ab is double. 
(2) The line a is an 8-tuple line: in fact, for a point on the line we have 
paa = 0, pba = 0, p 2 rySa. = 0, p 2 8a/3 = 0, p 2 a/3y = 0 ; and the factor vanishes, being = 0 1 . 
Each of the factors is 0 1 , and the norm is = 0 8 . 
39. (3) The line [ab, a, /3, 7] is a double line. To prove this, observe first that 
for a point on this line we have p 2 a(37 = 0. 
Taking as before z = 0, w = 0 for the equation of the line ab, a, /3, 7, we have 
h a = 0, hp = 0, h y — 0, and z a w b - z b w a = 0 ; or say w a = Az a , w b = Az b ; whence, writing for 
shortness / = — (g — Aa)x+ (/+ \b) y, viz. I a = — (g a — Aa a )x + (f a + h.b a )y, we have (when 
2 = 0, w = 0) pau = z a Ia, pba = z b I a , or omitting the factor Vz a z b , *Jpaa . pboi = / a ; and so 
for ^Ipafi.pbfi and 'Jpay.pby. The factor thus is 
I a • p 2 /3yS — 1^ . p 2 y8oc + I y . p 2 8a(3; 
viz. writing z = 0, w = 0 in the expressions of p 2 /3yS, &c., this may be written 
2 [(g -Aa)x-(f+ Xb) y] {(agh) x 2 + [(ahf) + (bgh)] xy + (bhf) y 2 }, 
where observe that £ denotes a sum of three terms of the form 
{Surface abcdafi.} 
a . /3yS — ¡3.7Sa + 7. Sa/3.