174 QUATERNIONS AND VECTOR ANALYSIS. 



evidently through inadvertence, and finding that the resulting 

 equations (thus interpreted) would not be true, he concludes that I 

 must have meant something else by the original equations. Now, 

 these equations will hold if interpreted in the quaternionic sense, as is, 

 indeed, a necessary consequence of their holding in the dyadic sense, 

 although the converse would not be true. My critic was thus led, 

 in consequence of the inadvertence mentioned, to suppose that I had 

 departed from my ordinary usage and my express definitions, and 

 had intended the products in these integrals to be taken in the 

 quaternionic sense. This is the sole ground for the last charge. 



The second charge evidently relates to the notations 3? s and $ x (see 

 Nature, vol. xlvii, p. 592). It is perfectly true that I have used a 

 scalar and a vector connected with the linear vector operator, which, 

 if combined, would form a quaternion. I have not thus combined 

 them. Perhaps Prof. Knott will say that since I use both of them it 

 matters little whether I combine them or not. If so I heartily agree 

 with him. 



The first charge is a little vague. I certainly admit that vectors 

 may be used in connection with and to represent rotations I have no 

 objection to calling them in such cases versorial. In that sense 

 Lagrange and Poinsot, for example, used versorial vectors. But what 

 has this to do with quaternions ? Certainly Lagrange and Poinsot 

 were not quaternionists. 



The passage in the major abstract in Nature which most distinctly 

 charges me with the use of the quaternion is that in which a certain 

 expression which I use is said to represent the quaternion operator 

 q( )q~ 1 (vol. xlvii, p. 592). It would be more accurate to say that 

 my expression and the quaternionic expression represent the same 

 operator. Does it follow that I have used a quaternion ? Not at alL 

 A quaternionic expression may represent a number. Does everyone 

 who uses any expression for that number use quaternions ? A 

 quaternionic expression may represent a vector. Does everyone who 

 uses any expression for that vector use quaternions ? A quaternionic 

 expression may represent a linear vector operator. If I use an 

 expression for that linear vector operator do I therefore use quater- 

 nions ? My critic is so anxious to prove that I use quaternions that 

 he uses arguments which would prove that quaternions were in 

 common use before Hamilton was born. 



So much for the alleged use of the quaternion in my pamphlet- 

 Let us now consider the faults and deficiencies which have been found 

 therein and attributed to the want of the quaternion. The most 

 serious criticism in this respect relates to certain integrating operators,, 

 which Prof. Tait unites with Prof. Knott in ridiculing. As definitions 

 are wearisome, I will illustrate the, use of the terms and notations 





