579 
with respect to the initial system; and conversely it is not difficult to 
prove that every rational point, line, or plane, in space, ts a net-point, 
net-line, or net-plane, whatever that initial system of five points 
may be. It follows that although no irrational point, line, or plane, 
can possibly belong to the net, with respect to which it zs thus 
irrational, yet it can be indefinitely approached to, by points, lines, or 
planes which do so belong: a remarkable and interesting theorem, 
which appears to have been first discovered by J/dbius ;* to whom in- 
deed, as has been already said, the conception of the net is due, but 
whose analysis differs essentially from that employed in the present 
Paper. 
[128.] As regards the passage from one net in space to another, let 
the quinary symbols of some five given points P, . . P;, whereof no four 
are in one plane, be with respect to the given initial system 4 . . the 
following :— 
P,=(@,--%,),--Ps= (Gs -- 05); 
and let a’. . e’ and wu’ be six coefficients, determined so as to satisfy the 
quinary equation [5.], 
a'(P,) + b'(P.) + e/(2s) + a’(2,) + e’(2s) = — U(U), 
or the five ordinary equations which it includes, namely, 
; A, 4+ as=..-= Ay, +..+ev,=—wU. 
Let 2’ be any sixth point of space, such that 
~ (®’) = aa’(P,) + yb! (22) + 20/(B,) + wd! (P,) + vel(Bs) + U(U); 
then this sixth point v’ can be derived from the five points P, . . P;, by the 
same ‘constructions, as those by which the point P = (xyzwv) is derived 
from the five given points ancpe.. For example, if we take the five 
points, 
A, = (10001), 8, = (01001), c, = (00101), », = (00011), z = (00001), 
we have the symbolic equation, 
(41) + (Bi) + (4) + @) - 8) = (); 
if then we write v’ =x +y+2z+w — 8v, the point (xyzwv’) is derived 
from A,B,C,D,E, by the same. constructions as (zyzwv) from ascpr. In 
Bene Se 
* See page 295 of the Barycentric Calculus. As regards the theory of homographic 
figures, chapter xxv. of the Géometrie Supérieure of M. Chasles may be consulted with 
advantage. But with respect to anharmonic ratio, generally, it must be remarked that 
Professor Mébius was thoroughly familiar with its theory and practice, when he published 
in 1827; although he called it by the longer but perhaps more expressive name of 
Doppelschnittsverhiiltniss (ratio bissectionalis). It may be added that he denotes by (a, 
C, B, D), what I write as (aBcD). 
