1902-3.] J. H. Maclagan-Wedderburn on Vector Functions. 411 
will be shown later that this admits of being expressed in a more 
■concise form. 
Putting U p = p in (15) we have 
T n+1 pSp n + T”pSp 4*n-iP 71-1 + • . • • + d — 0 , 
which shows that any vector in general meets the surface in (n + 1) 
points. Hence also the cone of asymptotic directions is 
S p4>p n = 0 . 
Differentiating (15) and putting dp = T3 — p, the equation of the 
tangent plane at any point p is 
S(ET- p)((n + l)i> n p n + n ^n-iP n ~ 1 . . ■ • ) = 0- 
'This can also be written S(C7 - p) V Fp = 0 . 
In this connection it is useful to note that since by (7) VF(p) = 
— %{n + l)<f>np n it is easy to pass in any particular case from Car- 
tesians to quaternions. 
As was indicated above, the general equation of the (w+l) th 
degree admits of being put in a more useful form than (15). If, 
•on the analogy of (2), we form the general homogeneous quaternion 
scalar expression of the n th degree, 
2H w+1 Sgr , 
where q is a constant and r a variable quaternion, it can obviously 
in the same way as before be put in the form 
(17) 
where <frr n is a quaternion function of r, possessing properties (3) 
to (6). In addition it may be useful to note that 
cfi7’ n = <fi(Vr + Sr) w , 
where the terms in the bracket may be expanded by the binomial 
theorem, as in (9) ; also if V refers to the constituents of r, 
V Sr</>r n = - {n + l)V<£r* + (n + 1)S </»*. V Sr ; 
ihence if V. Sr = 0 , 
So V S 'rcf>r n = — (n + 1)S crcf^r 11 . 
