158 QUATERNIONS IN THE ALGEBRA OF VECTORS. 



The quantities written SVo> and Ww, where o> denotes a vector having 

 values which vary in space, are of fundamental importance in physics. 

 In quaternions these are derived from the quaternion Vo> by selecting 

 respectively the scalar or the vector part. But the most simple and 

 elementary definitions of SVo> and Wo> arc quite independent of the 

 conception of a quaternion, and the quaternion Vo> is scarcely used 

 except in combination with the symbols S and V, expressed or 

 implied. There are a few formulae in which there is a trifling gain in 

 compactness in the use of the quaternion, but the gain is very trifling 

 so far as I have observed, and generally, it seems to me, at the expense 

 of perspicuity. 



These considerations are sufficient, I think, to show that the position 

 of the quaternionist is not the only one from which the subject of 

 vector analysis may be viewed, and that a method which would be 

 monstrous from one point of view, may be normal and inevitable from 

 another. 



Let us now pass to the subject of notations. I do not know 

 wherein the notations of my pamphlet have any special resemblance 

 to Grassmann's, although the point of view from which the pamphlet 

 was written is certainly much nearer to his than to Hamilton's. But 

 this a matter of minor consequence. It is more important to ask, 

 What are the requisites of a good notation for the purposes of vector 

 analysis ? There is no difference of opinion about the representation 

 of geometrical addition. When we come to functions having an 

 analogy to multiplication, the products of the lengths of two vectors 

 and the cosine of the angle which they include, from any point of 

 view except that of the quaternionist, seems more simple than the 

 same quantity taken negatively. Therefore we want a notation for 

 what is expressed by Sa/3, rather than Sa/3, in quaternions. Shall 

 the symbol denoting this function be a letter or some other sign ? 

 and shall it precede the vectors or be placed between them ? A little 

 reflection will show, I think, that while we must often have recourse 

 to letters to supplement the number of signs available for the ex- 

 pression of all kinds of operations, it is better that the symbols 

 expressing the most fundamental and frequently recurring operations 

 should not be letters, and that a sign between the vectors, and, as it 

 were, uniting them, is better than a sign before them in a case having 

 a formal analogy with multiplication. The case may be compared 

 with that of addition, for which a + /3 is evidently more convenient 

 than Z(a, /3) or 2a/3 would be. Similar considerations will apply to 

 the function written in quaternions Va/3. It would seem that we 

 obtain the ne plus ultra of simplicity and convenience, if we express 

 the two functions by uniting the vectors in each case with a sign 

 suggestive of multiplication. The particular forms of the signs which 



