435] 
ON THE SIX COORDINATES OF A LINE. 
97 
79. It is to be observed that the foregoing values give identically x+y+z+w= 1, 
so that the equation of the plane infinity is x + y + z + w = 0. The values of the 
coordinates {x, y, z, w) may be written 
x : y : z : w : 1 = PBGD : PGAD : PABD : PGBA : ABGD; 
or in the original form 
x : y : z : iv : 1 = PBGD : APCD : ABPD : ABGP : ABGD, 
as may be most convenient. 
80. Denoting the points (a, /3, 7, 8) and (a', /3', 7, S') by Q, Q' respectively, we 
have 
a : /3 : 7 : 8 : 1 = Q£CT) : ¿Q(7£ : ABQD : ABCQ : ABGD 
and 
a' : /3' : 7' : 8' : 1 = Q'£(7£ : ¿Q'(7£ : ABQ'D : ABCQ' : ABGD, 
and writing 
(a, b, c, f, g, h) = (/3f - /3' 7, 7a' - y'a, a(3' - a.'/3, aS' - a'8, /38' - £'8, 78' - 78), 
viz. the two sets being taken to be equal, a = (3f — ¡3'<y, &c. instead of merely pro 
portional, then it is easily seen that we obtain 
a : b : c : f : g : h : 1 
= AQQ'D : Q'BQD : QQ'GD : QBGQ' : ¿Q(7Q' : ABQQ' : ¿£(7£, 
that is, in order to form the first six combinations we successively replace 
(.B, G), (C, A), (A, B), (A, D), (B, D), (G, £) 
in ¿£(71) by (Q, Q'). 
Article No. 81. Resulting formula} of Transformation. 
81. For the transformation of coordinates if we assume 
x =(Ai, p 1, Vi, 2/o» • 2 'o> w 0 ), 
y =(\, P2, v 2 , P& » )> 
0 = (X 3 , /¿ 3 , i»„ p 3 $ „ )> 
W = (X 4) /a 4 , y 4 , p 4 $ „ ), 
and take also (a, b, c, f, g, h), (a 0 , b u , c 0 , f, g<> > 4>) respectively equal, instead of merely 
proportional, to the foregoing values, then, observing that for the point A 0 we have 
(x 0 , y 0 , z 0 , w 0 ) = (1, 0, 0, 0) we see that X 1} X 3 , X 3 , X 4 are the ABGD — coordinates of 
¿0 i and the like as to the other sets of coefficients; viz. we have 
X, 
x 3 
X 3 
x 4 
1 = A 0 BGD 
A A, CD 
ABAoD 
ABCAo 
ABGD 
Pi 
P‘2 
Pi 
Pi 
1 =£0 „ 
„ £0 » 
>> Do >) 
» D 0 
» 
V\ 
V2 
V3 
Vi 
1 == G 0 „ 
„ G 0 „ 
„ G 0 „ 
„ Co 
>> 
Pi 
P2 
Ps 
Pi 
II 
rH 
„ Bo „ 
„ Do,, : 
„ Do 
3? 
C. VII. 
13