94 Proceedings of Boyal Society of Pdinhw^gh. [jan. 6, 
unit-vector. Vectors, and unit-vectors, by their very definition, 
represent real lines, each of given direction and length, and a real 
quantity cannot be represented by an imaginary one. 
We define the symbol Up to be the operator by which we con- 
struct the length Tp, so as to give to that length the direction 
belonging to p. In expressing the vector p by the symbolical 
product 
p = Up X Tp , 
and in attributing reality to p, we are constrained to attribute reality 
to Up also. 
We may put the expression of p under the form 
p = (TJpxOx(^P), 
where I is supposed to represent the unit line : thus (Up x Z), or 
simply Up, represents a vector of unit length : hence the denomina- 
tion of unit-vector given to Up. 
Tor the proposed equation, we remark first that the 
symbolic square of the unit-vector 2 is not, properly speaking, a 
square in the algebraic sense, and that stands for the symbol SS 
which may be looked upon as a collocation arrived at in the process 
of the symbolic multiplication of two vectors, which contain the 
element S. We must therefore refrain from thinking that the 
element £ can be extracted from the symbol ££, by simply apply- 
ing to it an algebraical extraction of the square root. 
There exists a more extensive class of cases, in which we must 
avoid the introduction of the symbol - 1 in the place of unit- 
vectors, and for this end we shall propose the adoption of a certain 
principle which has its analogue in the principle of the inconverti- 
bility of the order of the factors in a vector product. 
We have, namely, for the versor 
q — cos w -f 2 sin , 
the result 
Vl 
= cos w-\-Z sin 10 , 
Tb 
where the fraction — is supposed to be reduced to its lowest terms, 
m 
n , 
w = —ti\ — N , 
m m 
and 
