VECTOR DIFFERENTIATION. 
77 
If the axis p is variable, then the reduction p 2 = 1 can be 
introduced only after the process of differentiation has been 
completed. To make the reduction before differentiation, is 
permissible only when the axis is constant. Upon this prin¬ 
ciple and that of external differentiation the following devel¬ 
opment rests. 
In Hamilton’s expression for p the axis is placed in the 
denominator and to the left of the symbol of differentiation; 
but according to Professor Tait’s definition in words, the axis 
belongs to the divisor element in the rate, and from the con¬ 
ventions already adopted the divisor element must be writ¬ 
ten after, not before, the dividend element. Let ( ) denote 
the place for the function, then 
■ _ n ) i , *( ) i , *( ) i 
dx i dy j dz k' 
In order to lay the foundation for a general method it is 
necessary to begin with a consideration of the elementary 
principles of the subject, and first it is necessary to find pi?, 
when i? — xi + yj -f zlc, in which i j h are constant. Now, 
A (xi + yj + zk) 4- = 1; 
4- (xi + yj + i = i; 
°y j 
-Z- C xi + yj + zk) -1 = 1, 
the unities obtained being absolute; that is, they are ob¬ 
tained by the absolute principle of reduction 4- = l,not by 
any such principle of final reduction as i l = 1. It follows 
that pi? = 3 absolutely. In Professor Tait’s Treatise on Qua¬ 
ternions we have, instead, — 3, and it is not an absolute 
number. The minus arises from imaginary vectors being 
used instead of real vectors, and its deduction involves the 
reduction i 2 = — 1. The 3, which here occurs, appears to be 
due to the tridimensional character of space. 
