Page:Scientific Papers of Josiah Willard Gibbs - Volume 2.djvu/189

 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 warn commendation 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,