QUATEKNIONS IN THE ALGEBRA OF VECTORS. 157 



our formulae to space of four or more dimensions. It will not be 

 claimed that the notions of quaternions will apply to such a space, 

 except indeed in such a limited and artificial manner as to rob them 

 of their value as a system of geometrical algebra. But vectors exist 

 in such a space, and there must be a vector analysis for such a space. 

 The notions of geometrical addition and the scalar product are 

 evidently applicable to such a space. As we cannot define the 

 direction of a vector in space of four or more dimensions by the con- 

 dition of perpendicularity to two given vectors, the definition of Va/3, 

 as given above, will not apply totidem verbis to space of four or more 

 dimensions. But a little change in the definition, which would make 

 no essential difference in three dimensions, would enable us to apply 

 the idea at once to space of any number of dimensions. 



These considerations are of a somewhat a priori nature. It may be 

 more convincing to consider the use actually made of the quaternion 

 as an instrument for the expression of spatial relations. The principal 

 use seems to be the derivation of the functions expressed by Sa/3 and 

 Va/3. Each of these expressions is regarded by quaternionic writers 

 as representing two distinct operations; first, the formation of the 

 product a/3, which is the quaternion, and then the taking out of this 

 quaternion the scalar or the vector part, as the case may be, this 

 second process being represented by the selective symbol, S or V. 

 This is, I suppose, the natural development of the subject in a treatise 

 on quaternions, where the chosen subject seems to require that we 

 should commence with the idea of a quaternion, or get there as soon 

 as possible, and then develop everything from that particular point of 

 view. In a system of vector analysis, in which the principle of de- 

 velopment is not thus predetermined, it seems to me contrary to good 

 method that the more simple and elementary notions should be defined 

 by means of those which are less so. 



The quaternion affords a convenient notation for rotations. The 

 notation q( )q~ l , where q is a quaternion and the operand is to be 

 written in the parenthesis, produces on all possible vectors just such 

 changes as a (finite) rotation of a solid body. Rotations may also be 

 represented, in a manner which seems to leave nothing to be desired, 

 by linear vector functions. Doubtless each method has advantages in 

 certain cases, or for certain purposes. But since nothing is more 

 simple than the definition of a linear vector function, while the 

 definition of a quaternion is far from simple, and since in any case 

 linear vector functions must be treated in a system of vector analysis, 

 capacity for representing rotations does not seem to me sufficient to 

 entitle the quaternion to a place among the fundamental and necessary 

 notions of a vector analysis. 



Another use of the quaternionic idea is associated with the symbol V. 



