288 PROFESSOR C. J. JOLY ON QUATERNIONS AND PROJECTIVE GEOMETRY. 
and on taking the scalar of both sides 
S.D2.(S.^-)" = 4n(n+ l)(S.^/)"-h 
From these results follow certain analogues of Laplace’s equation 
TD'Tr/”2 = 0, TDh/(D).T(r/ + «)-”“ = 0 .... (404); 
and 
S.D-(S.r/)--i = 0, S.Dh/(D).(S.((/ + a)-)-i = 0 . . . (405). 
Moreover, the general expression for the operator in terms of arbitrary differentials 
a, h, c, d of q enables us to write down a number of invariants and identities. For 
instance, operating on fq, we find 
l).f<l.{ahcd) = [bcd]fa — [acd]fh -\-[ahd]fc — [cdjc]fd . . (406). 
Other examples relating to integration will be found in a paper in ‘ Proc. Ptoy. Irish 
Acad.,’ vol. 24, Sect. A, pp. G- 20 . 
97 . So far as projective geometry is concerned, the use we make of the operator I) 
is to form successive polars of a point with respect to a surface and to show that it 
leads directly to AnoNHOLrfs notation. 
The polar of a point r with respect to a surface Q = 0 of order m is 
(SrD)".Q = 0 .(407). 
If n = m, the operator simply multiplies Q by a numerical factor and changes the 
quaternion involved from q to r. Thus we may write the equation of the surface in 
the form 
(SrI))'"Q = 0, or (Srn)“ = 0 .(408), 
where a is a symljolic quaternion devoid of meaning unless it enters into a term 
homogeneous in o. to the order m. This is equivalent to Aron wold’s method. 
.SECTION XVI. 
The Bilinear Quaternion Function. 
Art. Page 
98. Definition of the l.ilinear function /(/j.y). 289 
99. The permutate of a bilinear function f, iP'l) = f i'lp) > and permutable functions 
^{PQ) = -hf{P^j) + hfAPd . 289 
100 . Combinatorial bilinear functions C (/)._/) = b/(p'y) ~ p/, (p!/).289 
101. The first and second conjugates of a bilinear function. 289 
102. Successive conjugates and permutates. The six fundamental functions. 289 
103. The ^Mnvariants and the (/invariants. 290 
104. The first and second Jacobian surfaces I — J 0 = 0; the 7/ Jacobian corre¬ 
spondence, / {pq) = 0.290 
105. The second and third Jacobian correspondences,/'(;p) = 0 and/" Qu") = 0 . . . . 291 
106. The third Jacobian surface K (r) = 0. The 7A'and the KI Jacobian correspondences 291 
