X. 



QUATERNIONS AND VECTOR ANALYSIS. 



[Nature, vol. XLVIII. pp. 364-367, Aug. 17, 1893.] 



IN a paper by Prof. C. G. Knott on " Recent Innovations in Vector 

 Theory," of which an abstract has been given in Nature (vol. xlvii, 

 pp. 590-593; see also a minor abstract on p. 287), the doctrine that 

 the quaternion affords the only sufficient and proper basis for vector 

 analysis is maintained by arguments based so largely on the faults 

 and deficiencies which the author has found in my pamphlet, Elements 

 of Vector Analysis, as to give to such faults an importance which 

 they would not otherwise possess, and to make some reply from me 

 necessary, if I would not discredit the cause of non-quaternionic 

 vector analysis. Especially is this true in view of the warm com- 

 mendation and endorsement of the paper, by Prof. Tait, which 

 appeared in Nature somewhat earlier (p. 225). 



The charge which most requires a reply is expressed most distinctly 

 in the minor abstract, viz., "that in the development of his dyadic 

 notation, Prof. Gibbs, being forced to bring the quaternion in, logically 

 condemned his own position." This was incomprehensible to me until 

 I received the original paper, where I found the charge specified as 

 follows: "Although Gibbs gets over a good deal of ground without 

 the explicit recognition of the complete product, which is the difference 

 of his 'skew' and 'direct' products, yet even he recognises in plain 

 language the versorial character of a vector, brings in the quaternion 

 whose vector is the difference of a linear vector function and its 

 conjugate, and does not hesitate to use the accursed thing itself in 

 certain line, surface, and volume integrals" (Proc. R.S.E., Session 

 1892-3, p. 236). These three specifications I shall consider in their 

 inverse order, premising, however, that the epitheta ornantia are 

 entirely my critic's. 



The last charge is due entirely to an inadvertence. The integrals 

 referred to are those given at the close of the major abstract in 

 Nature (p. 593). My critic, in his original paper, states quite correctly 

 that, according to my definitions and notations, they should represent 

 dyadics. He multiplies them into a vector, introducing the vector 

 under the integral sign, as is perfectly proper, provided, of course, 

 that the vector is constant. But failing to observe this restriction, 



