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503] 
THE VERTICES OF CONES WHICH SATISFY SIX CONDITIONS. 
105 
nnulce. 
dng as centre of 
îrything upon an 
oints and lines in 
onic ; the equation 
e a conic passing 
me-coordinates are 
it represented by 
coordinates. The 
iven points a, &c., 
oned by so doing, 
iates of a line, it 
;ever to the given 
, b a , C a , fa, C/a> ha), 
lines ab, &c., but 
oining the vertex 
l the plane W — 0, 
'a = XW a X({W, &C.) , 
plane W = 0, the 
The following notations and formulae are convenient: 
9. pabc = 0 is the equation of the plane through the points a, b, c; viz. 
pabc = 
X , 
y » 
z , 
W 
x a, 
y a, 
z a> 
Wa 
Xj), 
yb, 
Zb, 
W b 
X c , 
yc 
Zc, 
w c 
Of course pbac=—pabc, &c. Observe that here, and in the notations which follow, 
the letter p is used as referring to the coordinates (x, y, z, w), and that the index 
of p (= 1 when no index is expressed) shows the degree in these coordinates. 
10. pad = 0 is the equation of the plane through the point a and the line a; 
viz. pact is the foregoing determinant, if for a moment b, c are any two points on 
the line a; or, what is the same thing, 
pa'x = P a x + Q a y + R a z + S a w, 
where 
P a = • hy a — gz a + ciw a , 
Qa == hx a . + fz a + bw a , 
Pa= gXa-fVa • +GW a , 
S a =- ax a - by a — cz a . ; 
and (a, b, c, f, g, h) are the coordinates of the line a: observe that paa=paa. 
11. p-oc^y = 0 is the equation of the quadric surface through the lines a, ¡3, 7; 
iz. we have 
p 2 a/3<y = (agh) x? + (bhf) y 2 + (cfg) z 2 + (abc) w 2 
+ [(abg) - (call)] xw 
+ [(bch ) - 06/)] yw 
+ [(caf) - (beg )] zw 
+ Wg) + W)J f 
+ [(cgh) + (afg)] zx 
+ [(ahf) + (bgh)] xy, 
here 
agh = 
a a, g a , h a 
a?, gp, K 
a y , g y > hy 
&G. 
.. K, c.,fg„ K), (a s ,...), (a y ....) being the coordinates of the given lines c, /3, y. 
bserve that p 2 /3ay — —p 2 a/3y, &c. 
C. VTTT 
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