VECTOR DIFFERENTIATION. 
BY 
Alexander Macfarlane. 
[Read before the Society March 31, 1900.] 
In 1895 I read a paper before the American Mathematical 
Society on Differentiation in Space Analysis * in which it was 
shown that there are two distinct kinds of differentiation, * 
and that only one of these is treated of in works on quater¬ 
nions. These two kinds of differentiation arise from the fact 
that in space analysis the order of the operations is in gen¬ 
eral material; for instance, the product of the square of each 
of two versors is different from the square of the product of 
the simple versors; so to differentiate the factors in situ gives 
a different result from differentiating the product as a whole. 
It is the first kind of differentiation which is discussed by 
Hamilton and Tait; it involves the peculiarity that there is 
in general no differential coefficient, and in consequence the 
generalized form of Taylor’s theorem is difficult to express. 
The second kind of differentiation yields a differential co¬ 
efficient, and Taylor’s theorem remains unaltered in form, 
only the order of the terms in the binomial must be pre¬ 
served in the terms of the expansion. Let AB denote the 
product of two vectors; the former kind of differentiation 
gives 
d(AB) = dA B + AdB; 
and for the square of the vector, 
d (A 2 ) = dA A + A dA. 
* Science, vol. i, page 302. 
11—Bull. Phil. Soc., Wash., Vol. 14. 
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