535 
although ¢his symbol represents xo determined point, because both the 
denominator and numerator of the expression (8) vanish, by (2) and (4), 
when the five coefficients xyzwv become each equal to unity. 
[6.] With these notations, if @ and 9’ be any other quinary symbols, 
and ¢ and u any two coefficients, we shall have the congruence, 
d=e, if e’=te+u; (17) 
the two points » and P’, which are denoted by these éwo symbols, in this 
case coinciding. Again the equation, 
Qe” =ta+ te + uv, (18) 
is found to express that @, 9’, @” are symbols of three collinear points ; 
” mw 
and the complanarity of four points, of which the symbols are a, 9’, @,” 2”, 
is expressed by this other equation of the same form, 
Qe” =te+ ta +t! e+ uv. (19) 
[7.] If then a variable point v be thus complanar with three given 
points, Po, Pi, Ps, its coordinates [4.] must be connected with theirs, by 
five equations of the form, 
L= bX + byt e+ Us..V=hy tty + fr. +u; (20) 
whence, by elimination of the four arbitrary coefficients t,t, t,u, a linear 
equation is obtained, of the form 
le+my +nz+rw+sv=0, (21) 
with the general relation 
l+m+ntr+s=0 (22) 
between its coefficients; and this equation (21) may be said to be the 
Quinary Equation of the Plane P,P? The five new coefficients murs 
may be called the Quinary Coordinates of that Plane; and the plane itself 
may be denoted by the Quinary Symbol, 
II=[J, m,n, r,s], or briefly, 1 =[7mnrs], (23) 
when the commas can be omitted without confusion. 
If 2, Rr’, . . be symbols of this form, for planes I, I’, .., then the 
equation 
R/=trR, (24) 
in which ¢ is an arbitrary coefficient, expresses that the two planes 
Tl, I’ coincide ; the equation 
R"=tr+t/R (25) 
~ expresses that the three planes I, I’, 1” are collinear, or that the third 
passes through the line of intersection of the other two; and the equation 
R”=tr+t Rr'+t'R! (26) 
expresses that the four planes I, I’, 11", 1” are compunctual (or concur- 
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