503] THE VERTICES OF CONES WHICH SATISFY SIX CONDITIONS. Ill 
(where 1 a = h a y — g a z 4- a a w = 0) or in the equivalent form wherein we have in the first 
term + instead of —, and in the second term the determinants 
Pa, 
Qa, 
Pa 
î 
Pa, 
Qa, 
Ra 
A , 
B , 
G 
A', 
B', 
G' 
A"\ 
B"\ 
G'" 
A", 
B", 
G" 
22. {The question, in fact, is to find the reciprocal of the form 
A (ax + by + cz) (a'x + b'y + c'z) - y (a"x + b"y + c"z) (a'"x + b"'y + c'"z) = 0 ; 
taking £, 7], Ç for the reciprocal variables, the coefficient of £ 2 is 
{A (be’ + b'c) - y (6V" + b'"c")Y - (2Abb' - 2yb"b"') (2Acc' - 2yc"c"), 
viz. this is 
A 2 (be - b'cf + y? (b"c'"- b"'c'J + 2\y [2bb'c’c'" + 2b"b'"cc'- (be' + b'c) (b"c" + b'"c")}, 
or, as it may be written, 
{A (be' - b'c) ± y (b"c'" - b'"c")Y + 2Ay ( 2bb'c"c'" + 2b"b'"cc \ 
+ (be - b'c) (b"c" - b’"c") 
-(bc' + b'c)(b"c'" + b'"c") 
. 
Taking the upper signs, this is 
[A (be' - b'c) + y (b"c'" - b'"c")Y + 4>\y I bb'c"c"' + b"b'"ccG 
\- bcb"c"' - b'cb"'c' 
viz. the term in Ay is 
= + 4Ay(bc"'-b'"c)(b'c"-b"c'). 
Taking the lower signs, it is 
{A (be - b'c) - y (b"c'" - b'"c")Y + 4A y / bb’c"c" + b"b'"cc'\ 
\- bcb"'c"-b'cb"c"T 
viz. the term in A y is 
4Ay(bc"-b"c)(b'c'"-b'"c)-, 
and it is thence easy to infer the forms of the other coefficients, and to obtain the 
reciprocal equation in the two equivalent forms 
£ , v , Ç } 2 + 4A/a 
a", b", c" 
a"\ V", c" 
V > Ç } 2 + 4A/a 
a", b", c" 
a'", b'", c"' 
which are the required auxiliary formulæ. 
{Surface cibcdea.) 
{A 
V, 
£ 
+ y 
a, 
b, 
c 
a', 
b', 
c' 
[A 
V, 
? 
a, 
b, 
c 
a', 
b', 
c' 
ç , v , Ï 
a , b , c 
a"', b'", c'" 
v > Ç 
a , b , c 
a", b", c" 
£ , V , K 
a', b' , c' 
a", b", c" 
Ç , v > Ç 
a', V , c' 
a'", b’", c'" 
= 0, 
= 0,