] 902-3.] J. H. Maclagan-Wedclerburn on Vector Functions. 409 
On the General Scalar Function of a Vector. By J. H. 
Maclagan-Wedderburn. Communicated by Dr W. Peddie. 
(MS. received 6th February 1903. Read March 2, 1903.) 
The general scalar function of a vector p may evidently be written 
in the form 
2 (^Sc^pSc^p . . . Sa n p) (1) 
n=l 
where a x . . . a n are constant vectors, each term being homogeneous 
in p. If the factors of each term of any one of the brackets be 
permuted in every possible way, i.e., in j ways (where r is the 
degree of the set of terms chosen), it can be written in the form 
s P^P r-1 (2) 
where </>p’' -1 is a vector function of p of the (r— l) th degree, the 
index (r — 1) serving merely to indicate the degree of the function. 
This notation will be found to lead to no confusion, and to have 
great advantages. Since the variable p occurs in </> along with each 
vector a, and also these vectors have been arranged in every possible 
way, it follows that, if in each term of <f> we replace s of the p’s 
by any other vector <r in such a way that the order of occurrence 
of the p’s and the cr’s is the same in each term, then the final result 
is independent of that order ; that is to say, 
<f>p n ~ a o* = <f><r s p n - s (3) 
It also immediately follows from the definition of <f> that 
So-^p 11 = Sp<£p n- V (4) 
Joly, who has investigated linear and vector functions of three 
variables, has called such a function completely self-conjugate (see 
Appendix to Joly’s edition of Hamilton’s Quaternions , page 467). 
The following useful properties follow immediately — 
d<f)p n = n(fjp n ~ 1 dp (5) 
dSp<frp? = (n + l)$dp<fip n (6) 
vSp</>p” = 1i{n + \)i$$p n dpjdx = -(n+l)<j>p n . . (7) 
SpVSp<£p n = — (n + l)3p</>p” (8) 
PROC. ROY. SOC. EDIK — VOL. XXIV. 27 
