A SIXTH MEMOIR UPON QUANTICS. 
589 
[158 
158] 
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221. As regards the analytical theory, suppose that the point-coordinates of the 
two points of the Absolute are (p, q, r), (p 0 , q 0 , r 0 ), then the line-equation of the 
Absolute is 
2 (pf + qv + rf) (poij + q 0 v + nK) = 0 ; 
so that we have gH = 2pp 0 , 23 = 2qq 0 , (& = 2rr 0 , $ = qr 0 + rq 0 , =rp 0 + pr 0 , ^=pq 0 +qpo, 
and thence K = 0 ; but 
K (a, b, c, f, g, h\x, y, zf = 
where obviously 
X , 
y > 
p > 
q> 
Po, 
q 0 , 
X , 
ih 
P> 
q> 
Po, 
Vo 
z 5 
r 
r 0 
z 
= 0 
r 
n 
is the equation of the Absolute line. 
222. The expression for the distance of the two points (x, y, z), {x, y', z') is 
given as the arc to an evanescent sine; but reducing the arc to its sine, and omitting 
the evanescent factor, the resulting expression is 
x, 
y> 
z 
X , 
y> 
z 
-4- 
X , 
y> 
z 
X , 
y» 
z’ 
x', 
y> 
/ 
Z 
X , 
y'’ 
z' 
p > 
q> 
r 
p > 
q > 
r 
p, 
q > 
r 
Po, 
qo, 
r 0 
Po, 
q 0 , 
n 
Po, 
qo, 
r 0 
and the expression for the distance of the two lines (f, y, £), (£', y, £') is 
cos _i (p% + qv + r 0 (po? + gov' + uO + (vZ + qv + r£') (pof + g 0 y + r£) 
V2 (p% + qy + r£) (p 0 f; + q 0 y + r£) V2 (p% + qy + r£") (p 0 f' + q 0 y' + r'£') 
or, what is the same thing, 
gin" 1 ^ r ° ~ rq °) ~ v '^ + ~ P r °) ~£v) + (Pgo ~ gPo) (£V ~ %'v) . 
V2 (p| + qy + r£) (p 0 £ + q 0 y + r 0 £) V2 (pf + qy' + r£) (p 0 ij' + q 0 y' + r 0 f') 
and finally, the expression for the distance of the point x, y, z from the line (£', y\ £'), 
reducing the arc to its sine and omitting the evanescent factor, is 
(ftf + yy + £z) -5- 
X , 
y> 
z 
p > 
q > 
r 
Po, 
qo, 
n 
V2 (pf' + qy' + rO (Pof + qoV + r <£')-