74 
MACFARLANE. 
The latter kind of differentiation gives 
d (. AB ) = A dB + B dA , 
and for the square, 
d ( A) 2 = 2 A dA. 
In the latter case the differentials dA and dB are supposed 
to be subsequent to both A and B, whereas in the former 
case dA is supposed to be prior to B. The former may be 
called internal , the latter external , differentiation. I do not 
say that one form of differentiation is true and the other 
false. I say that there are two distinct forms of differenti¬ 
ation. They differ in result because they differ in nature. 
In vector-differentiation it is the new form which is of prin¬ 
cipal importance, and in the present paper I shall indicate 
some of the developments which it yields immediately. 
Let S denote a vector, s its spherical modulus, and a its 
spherical axis; then, according to the new form of differen¬ 
tiation, n denoting any integral positive number, 
dS n = nS n ~ 1 dS ; or 
dS n 
dS 
= nS* 
ds n — ns n 1 ds , as usual; 
d(7 n = nt7 n ~ 1 d<r ; or 
and 
dS n = d (s n tf n ) ==ns u ~ 1 dsa n T ws n ~ 1 d<r. 
d<j n 
d(7 
If we write d <y n = ~ 1 d<r where the d<r is written after the 
(T n ~ \ it follows that the proper way to write the differential 
coefficient is not for the latter will differ from 
dff ’ dty 
the former by minus when n is odd. 
The symbol for vector differentiation is p, called nabla 
by Robertson Smith on account of its resemblance to an 
