533 
[2.| In general, if m denote five or any greater number of ¢ndepen- 
dent points of space, the number of the derived points of the form A * II, 
or AB‘CDE, which can be obtained by what is relatively to them a First 
Construction, of the kind just now described, is easily seen to be the 
function, 
so that f(5) = 10, as above, but f(15) =30080.. If then the fifteen points 
P,, P, were thus independent, or unconnected with each other, we might 
expect to find that the number of points P, derived from them, at the 
next stage, should eaceed thirty thousand. And although it was obvious 
that many reductions of this number must occur, on account of the 
dependence of the ten points P, on the five points P,, yet when I hap- 
pened to feel a curiosity, some time ago, to determine the precise number 
of those which have been above called Povnts of Second Construction, and 
to assign their chief geometrical relations to each other, and to the fifteen 
former points, it must be confessed that I thought myself about to un- 
dertake the solution of a rather formidable Problem. But the motive 
which had led me to-attack that problem, namely the desire to try the 
efficiency of a certain system of Quinary Symbols, for points, lines, and 
planes in space, which the Methed of Vectors had led me to invent, in- 
spired me with a hope, which I trust that the result of the attempt has 
not altogether failed to justify. And, in the present communication, I 
wish first to present some outline of what may be called perhaps a 
Quinary Calculus, before proceeding to give, in the second place, some 
sketch of the results of its application to the geometrical Wet in Space. 
Part 1.— On a Quinary Calculus for Space. 
[3.] Let ancpz be (as in [1.]) any five given points of space, 
whereof no four are situated in any common plane ; then, by decompos- 
ing Ep in the directions of Ba, EB, Ec, we can always obtain an equation 
of the form, 
a.FA+6.EB+¢C.EC+ad.ED=0, (1) 
in which the coefficients abed have determined ratios. And if we next 
introduce a fifth coefficient ¢, such that 
| at+b+e+d+e=0, (2) 
and add to (1) the identity 
(a+b+ce+d+e) on=0, (3) 
in which o is any arbitrary point (or origin of vectors), we arrive at the 
following equivalent but more symmetric form, 
@.o8+b6.0on+¢c.0c+d.0op+e.0E=0, (4) 
in which abede may be called the jive (numertcal) constants of the given 
