156 QUATERNIONS IN THE ALGEBRA OF VECTORS. 



representing in length the product of their lengths and the sine of 

 the angle which they include. This is denoted by Va/3 in quaternions. 

 How these notions are represented in my pamphlet is a question of 

 very subordinate consequence, which need not be considered at 

 present. The importance of these notions, and the importance of a 

 suitable notation for them, is not, I suppose, a matter on which there 

 is any difference of opinion. Another function of a and /3, called 

 their product and written a/3, is used in quaternions. In the general 

 case, this is neither a vector, like Va/3, nor a scalar (or ordinary 

 algebraic quantity), like Sa/3, but a quaternion that is, it is part 

 vector and part scalar. It may be defined by the equation 



The question arises, whether the quaternionic product can claim a 

 prominent and fundamental place in a system of vector analysis. It 

 certainly does not hold any such place among the fundamental 

 geometrical conceptions as the geometrical sum, the scalar product, 

 or the vector product. The geometrical sum a-\-/3 represents the 

 third side of a triangle as determined by the sides a and /3. Va/3 

 represents in magnitude the area of the parallelogram determined by 

 the sides a and /3, and in direction the normal to the plane of the 

 parallelogram. SyVa/3 represents the volume of the parallelepiped 

 determined by the edges a, /3 t and y. These conceptions are the very 

 foundations of geometry. 



We may arrive at the same conclusion from a somewhat narrower 

 but very practical point of view. It will hardly be denied that 

 sines and cosines play the leading parts in trigonometry. Now the 

 notations Va/5 and Sa/3 represent the sine and the cosine of the angle 

 included between a and /3, combined in each case with certain other 

 simple notions. But the sine and cosine combined with these 

 auxiliary notions are incomparably more amenable to analytical 

 transformation than the simple sine and cosine of trigonometry, 

 exactly as numerical quantities combined (as in algebra) with the 

 notion of positive or negative quality are incomparably more amenable 

 to analytical transformation than the simple numerical quantities of 

 arithmetic. 



I do not know of anything which can be urged in favor of the 

 quaternionic product of two vectors as a fundamental notion in 

 vector analysis, which does not appear trivial or artificial in com- 

 parison with the above considerations. The same is true of the 

 quaternionic quotient, and of the quaternion in general. 



How much more deeply rooted in the nature of things are the 

 functions Sa/3 and Va/3 than any which depend on the definition 

 of a quaternion, will appear in a strong light if we try to extend 



