366 



NA TURE 



[August 17, 189^ 



the subject (as they might almost b; said to give the mithema- 

 tic3 of the electro-magnetic field) I have written out more in 

 detail than might seem necessary. I have also called the atten- 

 tion of the student to many things, which perhaps he might be 

 left to himself to see. Prof. Knott says that the quaternionist 

 obtains similar equations by the simplest transformations. He 

 has failed to observe that the same is true in my Vector Analysis, 

 when once I have proved Poisson's Equation. Perhaps he takes 

 his model of brevity from Prof. Tait, who simplifies the sub- 

 ject, I believe, in his treatise on Quaternions, by taking this 

 theorem for granted. 



Nevertheless, since I am forced so often to disagree with 

 Prof. Knott, I am glad to agree with him when I can. He 

 says in his original paper (p. 226), "No finer argument in 

 favour of the real qua'ernion vector analysis can be found than 

 in the tangle and the jangle of sections 91 to 104 in the 

 'Elements of Vector Analysis.'" Now I am quite ready to 

 plead guilty to the tangle. The sections mentioned, as is suffi- 

 ciently evident to the reader, were written at two different 

 times, sections 102-104 being an addition after a couple of 

 years. The matter of these latter sections is not found in its 

 natural place, and the result is well enough characterised as a 

 tangle. It certainly does credit to the conscientious study 

 which Prof. Knott has given to my pamphlet, that he has dis- 

 covered that there is a violent dislocation of ideas just at this 

 point. For such a fault of composition I have no sufficient 

 excuse to offer, but I must protest against its being made the 

 ground of any broad conclusions in regard to the fundamental 

 importance of the qutternion. 



Prof. Knott next proceeds to criticise — or, at least, to ridicule 

 — my treatment of the linear vector function, with respect to 

 which we read in the abstract : — "As developed in the pam- 

 phlet, the theory of the dyadic goes over much the same ground 

 as is traversed in the last chapter of Kelland and Tait's ' Intro- 

 duciion to Qiiaternions.' With the exception of a few of those 

 lexicon pro lucts, for which Prof. Gibbs has such an affection, 

 there is nothing of real value added to our knowledge of the 

 linear vector function. " It would not, I think, be difficult to 

 show some inaccuracy in my critic's characterisation of the real 

 content of this part of my pamphlet. But as algebra is a formal 

 science, and as the whole discussion is concerning the best form , 

 of representing certain kin Is of relations, the important question 

 would seem to be whether there is anything of /yrwa/ value in 

 my treatment of the linear vector function. 



Now, Prof. Knott distinctly characterises in half a dozen 

 words the diffi;rence in the spirit and method of my treatment 

 of this subject from that which is traditional among quaternion- 

 ists, when he says of what I have called dyadics — "these are not 

 quantities, but opera'ors " (Nature, vol. xlvii. p. 592) I do not 

 think that I applied the word quantity to the dyadics, but Prof. 

 Knott recognised that I treated them as quantities — not, of 

 course, as the quantities of arithmetic, or of ordinary algebra, 

 but as quantities in the broader sense, in which, for example, 

 quaternions are called quantities. The fact that they may be 

 operators does not prevent this. Just as in grammar verbs may 

 be taken as substantives, viz. in the infinitive mood, so in algebra 

 operators — especially such as are capable of quantitative varia- 

 tion — may be regarded as quantities when they are made the 

 subject of algebraic comparison or operation. Now I would not 

 say that it is necessary to treat every kind of operator as quan- 

 tity, but I certainly think that one so important as the linear 

 vector operator, and one which lends itself so well to such 

 broader treatment, is worthy of it. Of cour -e, when vectors are 

 treated by the methods of ordinary algebra, linear vector 

 operators will naturally be treated by the same methods, but in 

 an algebra formed for the sake of expressing the relations be- 

 tween vectors, and in which vectors .are treated as multiple 

 quantities, it would seem an incongruity not to apply the methods 

 of multiple algebra also to the linear vector operator. 



The dyadic is practically the linear vector operator regarded 

 as quantity. More exactly it is the multiple quantity of the 

 ninth order which affords various operators according to the way 

 in which it is applied. I will not venture to say what ought to 

 be included in a treatise on quaternions, in which, of course, a 

 good many subjects would have claims prior to the linear vector 

 operator ; but for the purposes of my pamphlet, in which the 

 linear vector operator is one of the most important topics, I 

 cannot but regard a treatment like that in Hamilton's " Lec- 

 tures," or " Elements," as wholly inadequate cm the formal side. 

 To show what I mean, I have only to compare Hamilton's 



treatment of the quaternion and of the linear vector operator 

 with respect to notations. Since quaternions have been identi- 

 fied with matrices, while the linear vector operator evidently be- 

 longs to that class of multiple quantities, it seems unreasonable 

 to refuse to the one tho-;e notations which we grant to the 

 other. Thus, if the quaternionist has e^, logy, sin 17, cos <?, 

 why should not the vector analyst have <r*, log *, sin *, cos 4>, 

 whtre * represents a linear vector operator? I suppose the 

 latter are at least as useful to the physicist. I mention these 

 notations first, because here the analogy is most evident, liut 

 there are other cases far more important, because more ele- 

 mentary, in which the analogy is not so near the surface, and 

 therefore the difference in Hamilton's treatment of the two kinds 

 of multiple quantity not so evident. We have, for example, 

 the tensor of the quaternion, which has the important property 

 represented by the equation — '^{qr) = Tq'Xr. 



There is a scalar quantity related to the linear vector opera- 

 tor, which I have represented by the notation *| and called 

 the determinant of *. It is in fact the determinant of the 

 matrix by which * maybe represented, just as the square of the 

 tensor of y (sometimes called the norm ol q) is the determinant 

 of the matrix by which q may be represented. It may also be 

 defined as the product of the latent roots of *, just as the square 

 of the tensor of q might be defined as the product < f the latent 

 roots of q. Again, it has the property represented by the 

 equati )n 



I*-*! = KIlYi 



which corresponds exactly with the preceding equation with 

 both sides squared. 



There is another scalar quantity connected with the quater- 

 nion and represented by the notation Sq. It has the important 

 property expressed by the equation, 



S{qrs) = S{rsq) = S(.f./;-), 

 and so for products of any number of quaternions, in which the 

 cyclic order remains unchanged. In the theory of the linear 

 vector operator there is an important quantity which I have 

 represented by the notation *i, and which has the property 

 represented by the equation 



(■f^F-n^ = (T n-*)^ = (n-* 'P)s 



where the number of the factors is as before immaterial. *, may 

 be defined as the sum of the latent roots of *, just as zi-q may 

 be defined as the sum of the latent roots of?. 



The analogy of these notations may be further illustrated by 

 comparing the equations 



r(e„] = e^'7 

 and |.*1 = c^i- 



I do not see why it is not as reasonable for the vector analyst 

 to have notations like |*| and 4>^ as for the quaternionist (o have 

 the notations Tq and Sq. 



This is of course an argumenlnm ad qiiaternionislen. I do 

 not pretend that it gives the reason why I used these notations, 

 for the identification of the quaternion with a matrix was, I 

 think, unknown to me when I wrote my pamphlet. The red 

 justification of the notations |*] and ■t-^ is that they expreis 

 functions of the linear vector operator qnd quantity, which 

 physicists and others have continually occasion to use. And 

 this justification applies to other notations which may not hawt 

 their analogues in quaternions. Thus I have used *x to ex- 

 press a vector so important in the theory of the linear vector 

 operator, that it can hardly be neglected in any treatment of 

 the sulject. It is described, for example, in treatises »s 

 different as Thomson and Tait's Natural Philosophy and 

 Kelland and Tait's Quaternions. In the former treatise the 

 components of the vector are, of course, given in terms of the 

 elements of the linear vector operator, which is in accordance 

 with the method of the treatise. In the latter treatise the 

 vector is expressed by 



Voa' + V/3/3' -1- V77'. 



As this supposes the linear vector operator to be given not Iqr 

 a single letter, but by several vectors, it must be regarded as 

 eniirely inadequate by any one who wishes to treat the subject 

 in the spirit of multiple algebra, i.e. to use a single letter to 

 represent the linear vector operator. 



But my critic docs not like the notations |*1, ^^. *x- .His 

 ridicule, indeed, reaches high-water mark in the paragraphs m 

 which he mentions them. Concerning another notation, ♦x* 

 (defined in Nature, vol. xliii. p. 5'3^. he exclaims "Thus 



1 



NO. 



1242, vo:.. 48] 



