128 
ON THE SURFACES THE LOCI OF 
[503 
44. Investigation. In the projection, using line-coordinates, the equation of the 
conic touching the five lines may be written 
(f, v, O 2 
(P, Q, R) 2 
= 0; 
where the symbol denotes a determinant the last five lines of which are obtained by 
giving to (P, Q, R) the suffixes a, ¡3, y, 8, e respectively. This is at once transformed 
into 
a/3e. y8e . ay A . S/3 A — aye . S/3e. a/3 A . ySA = 0, 
or, what is the same thing, 
or say 
p 2 a/3e. p 2 y8e. ay A. 8/3 A — p 2 a ye .p 2 8/3e . a/3 A. ySA = 0 ; 
p 2 a/3e .p 2 ySe (A"£ + B"v + C"Z) (A"'£ + B'"y + G"'0 
— p 2 aye. p 2 8/3e (A£ + Br/ + C£) (A'(j + By + G'£) = 0 ; 
where p 2 a/3e, &c., signify as before ; and 
A£ + B V + CZ = 
z > 
V , 
K 
Pa, 
Qa, 
Ra 
P(>, 
Qp> 
Rp 
and so for A'% + B'y + C'£, &c., the suffixes for A', B', C being (7, 8); and those for 
A"t; + B"y + C"£ and A'"% + B'"y + C”'% being (a, 7) and (8, /3) respectively. 
45. Passing to the reciprocal equation, and making the conic pass through the 
point a, we obtain the equation of the surface in the form 
{p 2 aye .p 2 8/3e 
Pa, Qa, r a 
— p 2 a[3e . p 2 ySe 
Pa , Qa , r a 
A, B, G 
A", B", G" 
A', B\ C' 
A"', B"', G'" 
+ kp 2 aye ,p 2 S/3e .p 2 a/3e ,p 2 ySe 
Pa , Qa , 1'a 
Pa , Qa , r a 
A , B , G 
A', Bf , O' 
A", B", G" 
A'", B"', C" 
or in the equivalent form, where in the first term we have + instead of —, and 
in the second term the determinants are 
Pa , 
Qa , 
r a 
? 
Pa , 
Qa, 
r a 
A , 
B , 
G 
A', 
B\ 
G' 
A"', 
B"', 
G m 
A", 
B", 
C" 
46. To reduce this result, observe that we have 
A, B, G = 
hy — gz + aw, 
h'y — gz + a'w, 
— hx +fz+bw, gx —fy + cw 
— h'x + f'z + b'w, g'x — fy + cw 
{Surface aafi^8e.\