412 Proceedings of Royal Society of Edinburgh. [sess. 
It will now be shown that if Sr is constant, say equal to unity, 
and Vr = p , then (17) is the general scalar expression of the- 
(n-r l) th degree in p. For, since p can be expressed in terms of 
any three non-coplanar vectors, it may be easily shown that the 
general expression of the (n + 1 ) th degree involves 
(n + 2)(w + S)(n + 4) 
“ 31 
independent constants, but this is also the number of independent 
constants involved in (17), for it requires in general multiples of 
four independent quaternions to express any other quaternion.. 
(See Salmon, Geometry of Three Dimensions , page 233.) 
If the restriction that Sr = 1 be removed, then (17) represents 
what Hamilton has termed a full surface. 
The following are a few illustrations of the use of this form of 
the general equation to a surface. 
To find the tangent plane to Sr<£r n = 0 at the point Vr, we 
have 
S dr<f>r n = 0 
S(p-r)<£r w = 0 
hence Sp<£r 1? = 0 
where Sp = Sr = l, and Yp is the vector of any point in the- 
tangent plane. 
Differentiating a second time, we get 
%S<ip</>r n-1 <ir + S d 2 r(f>r n = 0 . 
If <f>r n = 0 for any value of r, the point is a conical point,, 
hence 
S dr$r n ~ x dr = 0 , 
therefore the equation to the tangent cone at the conical point is 
Sp<£r M-1 p = 0 . 
Similarly for an s-ple point we have 
Sp<£r ,l ~ s+1 p s_1 = 0 . 
Since the above was in print, I have found that equation (11) has 
been given by Tait (3rd ed., p. 420, Ex. 34) for the case r = n+ 1.. 
See also Kimura, Annals of Mathematics , Yol X., 1896, p. 127. 
(. Issued separately June 5, 1903.) 
