352 ON A CASE OF THE INVOLUTION AFA BG+ CH= 0. [939 
where C is a quadric function to be determined; or, what is the same thing, we have 
(sx 2 + qy 2 + z 2 — pyz — rxy ) (s'x 2 + q'y 2 + z 2 — p'yz — r'xy ), 
— (s"x l + q"y 2 + z 2 — p"yz — r'xy) (s'"a? + q"y 2 + z 2 — p"'yz — r"xy), 
+ (y 2 — zx) C= 0. 
Writing for shortness 
0i + 6, = a , 6,6, = /3 , 
6 3 + 6 X = a! , 6,6 4 = & , 
6 5 + 6 6 =ol", 6A = P", 
6 7 + 6 s = a"', 6 v 6 s =r, 
we have 
p = a + a 
p' = a " + 
p" = « + a" 
/" = a' + a'" 
q = aa' + /3 + /3' 
= a"a'" + /3" + /3"' 
5'" = «a" + /3 + /3" 
g'" = a'a'" + a '/8'" + a'"/3' 
r — a/3' A a'/3 
r' = a"/3'" + «'"/3" 
r" = a /3" + a"/3 
r"' = a.'/3'" + a"/3' 
*=/3/3' 
s' = /8"/8"' 
II 
s'" = /3'/3'". 
In the last-mentioned equation, the first and second lines together are a quartic 
function of (x, y, z), say the value is 
= Ax? + By 4 + Gz 4 , 
+ Fy 3 z + Gz 3 x + Hx?y, 
+ Iyz 3 + Jzoc? + Kxy 3 , 
+ Lx 2 yz + Mxy 2 z + Nxyz 2 , 
+ Py 2 z 2 + Qz 2 x 2 + Rx 2 y 2 , 
where after all reductions 
A= ss'-s's'" 
B = qq' -q"q" 
C = 1-1 
= 0, 
= (a/3'"-a"'/3) (a'-a") 
+ (ct'/3" - a"A) (« - a'") - (/3' - /3") ((3 - ¡3"'), 
= 0, 
F = — pq — p'q + p"q" A p'"q" 
G = 0-0 
H = — rs' — r's + r"s" + r"s" 
= (« - O - £") + («' - «") (¡3 - r\ 
— 0, 
= 0, 
I = —p—p' A p" Ap'" =0, 
J = 0-0 =0, 
K = — qr' — q'r A q'Y" + q'"r" 
L = - ps - p's + p"s'" A p'"s" 
M — pr + p'r — p"r'" — p'"r" 
N = — r — r + r" A r"' 
P = pp +q + q' - - q" - <i" 
= («/3 W - «"73) (/3" - /S') + («73" - «"/3') (/3"' - /8), 
= (a/3"' - «'"/3) (/3' - /3") + («'/3" - a"/3') (/3 - ¡3'"), 
= (off" - *'"(3) (a" - a') + (a'/3" - «"/S') («'" - a), 
= (a - a"') (/3" - /80 + («' - «") (/3"' - /8), 
= 0, 
Q = s + s'-s"- s'" = (A' - /3") (/3 - /3"'), 
= rr' + qs' + qs — r"r" — q"s"' — q"s" — 0 :