534 
system of five points, A.., although only their ratios are important, 
and (as above) their swm is zero. 
[4.] Let » be any other point of space, and let axyzwv be coefficients 
satisfying the equation, 
(w7—0v)a.PA+(y—v)b.PpB+(s—v)¢.Pc+(w-—v)d.Pp=0; (5) 
then, adding the identity, 
v(a.pat+b.pp+¢.pc+d.pp+e¢.PE) =0, (6) 
which results from (4), we obtain this other symmetric formula, 
za.PA+ Yb. PB+%¢.PC+ wd. PD+ve. PE= 0, (7) 
which may also be thus written, 
2a.0A + Yb. OB+ 8¢.0C+ wd.0D+ ve. 0B 
ai ’ (8) 
20 + yb + 2¢ + wd + ve 
o being again an arbitrary origin; and the five new and variable coeffi- 
cients, xyzwv, whereof the ratios of the differences determine the position 
of the point », when the five points a..£ are given, may be. called the 
Quinary Coordinates of that Point P, with respect to the given system of 
five points. 
[5.] Under these conditions, we may agree to write, briefly, 
P=(a, y, 8, W,V), or even P=(ayzwv), (9) 
whenever it seems that the omission of the commas will not give rise to 
any confusion; and may call this form a Quinary Symbol of the Point v. 
But because (as above) only the ratios of the differences of the coefficients 
or coordinates are important, we may establish the following Formula 
of Quinary Congruence, between two equivalent Symbols of one common 
point , 
(x! y's! w! v') = (ayzwv), (10). 
if 2'-v': y'-v':8-v:w-v=a-v:y-v:8s—v:w-v; (11) 
reserving the Quinary Equation, 
(a' y’ 3! w' v') = (ayzwv), (12) 
to imply the coexistence of the five separate and ordinary equations, 
w=2,y sy, 3 =%, w' =w, v=. (18) 
We shall also adopt, as abridgments of notation, the formule, 
t (a, y, 2, w, ¥) = (ta, ty, tz, tw, tv); (14) 
(a! ..0’)+ (@..0) = (a +2, .. 0/40); - (15) 
and shall find it convenient to employ occastoueay what ae be called 
the Quinary Unit Symbol, 
v=(11111); - (16): 
