ON THE SIX COORDINATES OF A LINE. 
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coordinates x, y, z, of any point on the line OQi); and then the condition for the 
pair of tractors may be written 
p 2 a 2 + p 3 a s + PiU4 + p 5 a s = 0, 
pJh +p-A + p A b A +p 5 b 5 = 0, 
p 2 c 2 +p 3 c s + P^ +p 5 c s = 0, 
P>fi +P3/3 +1^/4 +p 5 f 5 = F lt 
p 2 g 2 +P393 +Pi9i +p 5 g a = Gi, 
pJh +p 3 h 3 + pA + pA = 
where p 2 , p 3 ... are arbitrary coefficients; and we hence deduce 
pF x + (tG 1 + tH x = 0; 
but in precisely the same way, if the line 0Q 2 have with the lines 1, 3, 4, 5, a pair 
of tractors, and if F 2 , G 2 , ff 2 , be the coordinates of a point on the line OQ 2 , and 
similarly for the lines OQ 3 , OQ it 0Q b , and the coordinates (F 3 , G 3 , H 3 ), (F 4 , 6r 4 , H A ) 
(F 5 , G 5 , H & ), we have 
pF 2 + gG 2 + tH 2 = 0, 
pF 3 + o~G 3 + 7H 3 = 0, 
pF A + (tG a + tH a = 0, 
P F 5 + <jG 5 + tH 5 = 0, 
and these equations show that the five lines OQ lt 0Q 2 , 0Q 3 , OQ it 0Q 3) lie in the plane 
px + cry + tz = 0 ; 
so that this plane is given as the plane through the lines 0Q 1 , OQ 2 , 0Q S , OQ A , OQp, 
and we have thus (given the lines 1, 2, 3, 4, 5, and the arbitrary point 0) the con 
struction of the line (a, b, c, f g, h) through 0 in involution with the given lines. 
50. The original onefold relation may be replaced by the six equations 
Xu + AjCq + X 2 a 2 + A 3 a 3 + X 4 a 4 + X 5 u 5 = 0, 
\b + xA + \ 2 b 2 -t- XA + x 4 6 4 + x 5 Z> 5 = 0, 
X/i -f- X x h 2 + X 2 /?2 + X3J13 + XA + X 5 h5 — 0, 
and hence denoting as before the moments by 01, 02, 12, &c. we have 
. XjOl + A 3 02 + A 3 03 + A 4 04 + X 5 05 = 0, 
X10 . 4- X 3 12 -f- X 3 13 4- A 4 14 4- X 5 15 =0, 
X20 4- Xj21 . + X 3 23 + A 4 24 + X 5 25 = 0, 
X30 + Xj 31 + X 2 32 . 4- X 4 34 4- X 5 35 = 0, 
X40 4* Xj 41 + X 2 42 -f- X 3 43 . + A 5 45 = 0, 
XoO -f Xj51 + X 2 o2 -f- X 3 53 -1- X 4 54 . = 0,