538 
“The Quotient of any two given homogeneous and linear Functions, of 
the Differences of the Quinary Coordinates of a variable Point in Space, 
can always be expressed as the Anharmonic of a Pencil of Planes, whereof 
three are gwen, while the fourth passes through the variable Point, and 
through a given Right Line, which is common to the three former Planes.” 
[12.] For example, we find thus that 
xv y-v a) ! 
gt (Bc . AEDP); rca (CA. BEDP) ; Pom (AB.cCEDP); (42) 
and that 
fae = (CD. AEBP) ; i = (AD. BECP) ; —— =(BD.CEAP); (48) 
the product of these three last anharmonics of pencils being therefore 
equal to positive unity, so that we haye, for any six points of space, 
ABCDEF, the general equation, 
(AD. BECF) . (BD. CEAF) . (cD. AEBF) = 1. (44) 
If then we suppress the fifth coefficient, v, in the quinary symbol (9) of a 
pownt P, which comes to first substituting, as the congruence ( 10) permits, 
the differences z—v, y—v, 3—v, w—v, and v— vor 0, for a, y, 2, w, and v, 
and then writing simply @, . . w instead of z—v,.. w—v, and omitting the 
final zero, whereby the quinary symbol (00001) for the fifth given point 
# (27) becomes first (-1,-1,-1, -1, 0), or (11110), and is then reduced 
to the quaternary unit symbol (1111), we shall fall back on that system of 
anharmonic coordinates in space, of which some account was given in a 
former communication* to this Academy: the anharmonic (or quater- 
nary) symbol of a plane Il being, in like manner, derived from the qui- 
nary symbol (23), by simply suppressing the fifth coefficient, or coordi- 
nate, s. Anharmonic coordinates, whether for point or for plane, are 
therefore included in quinary ones ; but although they have some advan- 
tages of semplicity, it appears that their less perfect symmetry, of reference 
to the five given points a .. B, renders them less adapted to investigations 
respecting the Geometrical Net in Space, which is constructed with those 
Jwe points as data: and that therefore they are less fit than guinary co- 
ordinates for the purposes of the present paper. 
[13.] Retaining then the guinary form, we may next observe that al- 
though, when the five coefficients l..s are given, asin [7.], and the coordi- 
nates z..v of a point pare variable, the linear equation lx +. . + sv =0 (21) 
may be said to be the Local Equation of a Plane, namely of the plane 
[2..8], considered as the locus of the point (x..v); yet if, on the con- 
trary, we now regard #..v as given, and J. .s as variable, the same linear 
* See the Proceedings for the Session of 1859-60. 
