228 PKOFESSOR C. J. JOLY ON QUATERNIONS AND PROJECTIVE GEOMETRY. 
The symbol of the plane containing three points a, h, c, may be written in the 
form 
2) = [a,h,c] .(10); 
and on transformation this becomes 
njy, =-- [fajbjc] = F' [ahc] = F'p =-- F'f'p^ .(n), 
where n is a certain scalar and rvhere F' is an auxiliary function. 
In fact, the first equation sums up the last article ; in the second a new lunction 
F' is introduced, and in the fourth equation (9) is utilized. 
Since is quite arbitrary (11) may be replaced by the symbolical equations 
n = F'f ; F' = nf -^; f = rr^F'-'^ ; n = f'F' . . . . (12), 
an arbitrary quaternion being understood as the subject of the operations. 
Moreover, because 
n^pq = ^pF'fq = ^Fpfq:=^qfFp .( 13 ), 
where j? and q are arbitrary quaternions and where F is the conjugate of F\ it 
appears that 
u =/i^; F = nf -^; f= n'^F-^; n = Ff .( 14 ). 
And for any three arbitrary quaternions 
F[ahc] = [f'af'bf'c'] .( 15 ) 
as appears from symmetry, or, anew geometrically, Ijy considering a point as the 
intersection of three planes. 
Operating on the last equation Ity Sf'd we find, since 7i =fF —f'F’, 
n{abcd) = ifnf'bf'cf'd) ={fafbfcfd) .( 16 ). 
I he fact that («/;cd) is a combinatorial function of o, b, c and d proves that ii is an 
invariant, or that it is quite independent of any particular set of quaternions, or, b, c, d. 
This invariance is, however, established by the form of the equations (12) and ( 14 ). 
6. Keplacing / by// =J—t, where t is an arbitrary scalar, Hamilton denotes by 
/q and ii/ the auxiliary function and the invariant which bear the same relations to 
Jt that F and n bear to f. 
By ( 15 ) and (IG), Ft and iq are of the forms 
Ft = F-tG + dII-d‘ .( 17 ), 
where G and II are new auxiliary functions ; and 
lit = n — til + thi" — dll!” + .( 18 ), 
where rd, ii!' and ii'” are new invariants. 
