580 
particular, p is related to a,B,c,D,E, as the point p = (00081) is re- 
lated: to ancpE; but this point P satisfies the anharmonic equation, 
(pD,EP) = + 3; if then B, = D,E~ 4,B,c, = (00012), we must have the cor- 
_ responding equation (D,E,ED) = + 8: which is accordingly found to exist, 
and furnishes a construction for exscribing a pyramid acy to @ given 
pyramid A,B,0,D,, with which it is to be homologous, and to have a given 
point © for the centre of their homology, agreeing with the construc- 
tion assigned in [126.] for a similar problem of exscription. And in 
general, from any fie given points of a net, whereof no four are com- 
planar, we can (as was first shown by Mébius) return, by linear construc- 
tions, to the five initial points a... £; and therefore can, in this way, re- 
construct the net. 
[129.] If we content ourselves with quaternary (or anharmonic) 
coordinates [12.], or suppose (as we may) that v = 0, the equation of a 
surface of the second order takes the form, 
0 = f(ayzw) = ax? + By? + y2? + bw? 
+ Weye + Sze + nay) + Qw (Ox + wy + Kz); 
and if the ten coefficients a . . «, or their ratios, be determined by the 
condition that the surface shall pass through nine given net-points, those 
coefficients may then be replaced by whole numbers, and the surface may 
be said to be rationally related to the given net, or to the initial system 
A ..£, or briefly to be (comp. [8.]) a Rational Surface. For example, 
if the nine points be aBcpEc'a’c,A,, so that, besides passing through x, 
the surface has the gauche quadrilateral ancy superscribed upon it, the 
equation is 
L..0=f= a2 - yw; 
and if they be a, B, A’, B’, A», By, A,, A = (1210), andr=(1201), so that 
this new point r, like a“, belongs to the group P,,,, the equation of the 
surface is then found to be, 
Il..0=f=w+2-(wt+z) (a@+y) - 2axy. 
[180.] In general, whether the surface of the second order be ra- 
tional or not, it results from the principles of a former communication 
that any point Pp = (xyzw) of space is the pole of the plane I= [XYZW ], 
if X.. Whe the derwatives, 
X=0n,f, Y=n,f, Z=0d.f, W = duf3 
hence, in particular, the pole of the plane [x] of homology of the three 
pyramids «, 2’, B', [26.][113.] [125.], of which plane the quaternary 
symbol [12.] is [1111], is the point x determined by the equations, 
X= Y=Z= W, or vf =d,f= df = Daf; 
and if the point = be the mean point of the pyramid azcp, the plane [2] 
is then infinitely distant, and this point x is the centre of the surface. 
