a 2 (b 2 + & + d 2 ) (b 2 + c 2 ) p 2 + &c. 
The equation :£) 2 F=0, is then to be satisfied independently of the values of 
(-d., B, G, D) and (a, b, c, d), and as £> contains 16 distinct terms, ID 2 will contain in 
all 216.17 or 136 distinct terms. The equation ^ 2 F=0 gives therefore a plexus of 
136 equations, and the equations in each succeeding plexus, involved in £> 3 F=0, 
£> 4 F=0, &c. will, of course, be still be more numerous. 
If V = 0 be the plane conic which is the intersection of the surfaces 
X 2 _|_ y2 w 2 _ 
ax 4- bij 4- cz + dw = 0, 
then we have 
b 2 + c 2 , 
— ab , 
— ac , 
• » 
cd , 
— bd 
) (p, q, r, s, t, u) 2 . 
— ba, 
c 2 + ft 2 , 
— be , 
— cd , 
• y 
ad 
— ca, 
— cb , 
a 2 + b 2 , 
bd , 
— ad , 
• ? 
— cd , 
bd , 
ft 2 4 d 2 , 
ab , 
ac 
cd, 
. , 
— ad , 
ba , 
b 2 4* d 2 , 
be 
— bd, 
ad , 
• > 
ca , 
cb , 
c 2 4- d 2 
The values of P, Q, R, S, T, U (omitting a common factor 2) are 
P = (b 2 4- c 2 , — ab, —ac,., 4- cd, — bd) (p, q, r, s, t, u), 
&c., 
and if we proceed to form a term in D 2 F, say the coefficient of d. 2 a 2 , this is 
(Ud r + Td q + Pd„Y V, or 
(a 2 + b-, c- + a 2 , ft 2 + d 2 , — cd, + bd, — cb) ( U, T, P) 2 . 
The coefficient therein of p 2 is 
(ft 2 + b 2 , c 2 + ft 2 , ft 2 + d 2 , - cd, + bd, — cb) (— bd, cd, b 2 + c 2 ) 2 , 
that is, it is 
(ft 2 + b 2 ) b 2 d 2 — 2cd. cd (b 2 + c 2 ) 
+ (c 2 + ft 2 ) c 2 d 2 + 2bd. —bd(b 2 + c 2 ) 
+ (ft 2 + d 2 ) (b 2 + c 2 ) 2 — 2cb . — bd. cd, 
where the terms in which (6 2 + c 2 ) does not appear as a factor are together equal to 
a 2 d 2 (b 2 + c 2 ) + d 2 (b 2 + c 2 ) 2 , 
the entire expression thus divides by b 2 + c 2 , the quotient being 
’ (ft 2 + d 2 ) (b 2 + c 2 ) — 2c 2 d 2 — 2b 2 d 2 + a 2 d 2 + d 2 (b 2 + c 2 ), 
which is equal to a 2 (b 2 + c 2 + d 2 ), or restoring the factor b 2 + c 2 , we see that in !D 2 F 
the coefficient of ^l 2 a 2 is