224 PROFESSOR C. J. JOEY ON QUATERNIONS AND PROJECTIVE GEOMETRY. 
Again Hamilton writes 
[a, b,c] = {a,h,c)- [6,c]Sa - [c,a]S6 - [a, 6]Sc; (a, b,c) = 8[a, 6, c] = SYaV&Vc . (G); 
or if we replace a, 6, c by (1 + a)Sa, (1 + /3)S6, (I + y)Sc, where a, j3 and y are the 
vectors from the scalar point to three points a, b and c, we have 
[a,B,c] = SaySy - V(^y+ ya + ay8); (a, B, c) = Sa/3y . . . (H). 
Hence it appears that [a, b, c] is the symbol of the plane a, b, c; for 
— V \_a, b, c] (a, b, c)"^ is the reciprocal of the vector perpendicular from the scalar 
point on that plane. Also (a, b, c) is the sextupled volume of the tetrahedron oabc. 
Again, Hamilton writes for four quaternions 
(abed) = S . a\hcd^ . (I) ; 
and in terms of the vectors this is seen to be the products of the weights into the 
sextupled volume of the pyramid (abcd). 
Other notations may ol course he employed for these five combinatorial functions 
of two, three, or four quaternions or points, but Hamilton’s use of the brackets seems 
to 1)6 quite satisfactory. 
In the same article Hamilton gives two most useful identities connecting any five 
quaternions. These are 
and 
a (bode) + b{cdea) + c{deab) + d{eabc) + e(abcd) = 0. 
e{abcd) = [bcd]Sae — [accf\Sbe + [a6c^]Sce — [a6c]Sde 
(J) , 
(K) , 
which enable us to express any point in terms of any four given points, or in terms 
of any four given planes. 
The equation of a plane may be written in the form 
= 0.(L) ; 
and thus I, any quaternion whatever, may he regarded as the symbol of a plane as 
well as of a point. 
Gn the whole, it seems most convenient to take as the auxiliary quadric the sphere 
of unit radius 
.(M), 
wliose centre is the scalar point. With this convention the plane SIq = 0 is the 
polar of the point I with respect to the auxiliary quadric ; or the plane is the 
reciprocal of the point 1. Thus the principle of duality occupies a prominent 
position. 
The formulae of reci 2 )rocation 
{[abc] ; [abd]) = [ab] {abed); [ [abc] ; [a?)d] ] = — (ab) (abed ). . . (N) 
connecting any four quaternions are worthy of notice, and are easily proved by 
