August 17, iJ^93] 



NA TURE 



365 



vectors may be used in connection with and to represent rota- 

 tions. I have no objection to calling them in such cases versorial. 

 In that sense Lagrange and Poinsot, for example, used versorial 

 vectors. 15ut vf hat 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 ^aid to represent the 

 quaternion operator (/( ),/-i (vol. xlvii. p. 592). It would be 

 more accurate to say that my expression and the quaternionic 

 expression represent the same operator. Doe^ it follow that I 

 have used a quaternion ? Not at all. A quaternionic expression 

 may represent a number. Does everyone who uses any ex- 

 pression for that number use quaternions ? K 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 

 quaternions ? 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 weaiisome, 

 I will illustrate the use of the lernis and notations which I have 

 used by quoting a sentence addressed to the British Association 

 a few years ago. The speaker was Lord Ivelvin. 



"Helmholtz first solved the problem — Given the spin in any 

 case of liquid motion, to find the mo ion. His solution consists 

 in finding the potentials of three ideal distributions of gravita- 

 tional matter having densities respectively equal to i/t of the 

 rectangular components of the given spin ; and, regarding for a 

 moment these potentials as rectangular components of velocity 

 in a case of liquid motion, taking the spin in this motion as the 

 velocity in the required motion " (Nature, vol. xxxviii. p. 569). 

 In the terms and notations of my pamphlet the problem and 

 solution may be thus expressed : 



Given the curl in any case of liquid motion— to find the 

 motion. 



The required velocity is 1/41 of the curl of the potential of 

 the given curl. 



Or, more briefly— The required velocity is L of the La- 



4ir 

 placian of the given curl. 



Or in purely a'nalytical form— Required a. in terms of V x a 

 when V. w = o. * 



Solution— 



« = l/4irv X Pot V X w = l/4ir Lap V x a. 



(The Laplacian expresses the result of an operation like that 

 by which magnetic force is calculated from electric currents dis- 

 tributed in space. This corresponds to the second form in 

 which Helmholtz expressed his result.) 



To show the incredible rashness of my critics, I will remark 

 that these equations are among those of which it is said in the 

 original paper (Proc. R..S.E., Session 1892-93, p. 225), "Gibbs 

 gives a good many equations— theorems I suppose they ape at 

 being. ■ I may add that others of the equations thus charac- 

 terised are associated with names not less distinguished than 

 that of Helmholtz. But that to which I wish especially to call 

 attention is that the terms and notations in question express 

 exactly the notions which physicists want to use. 



But we are told (Natuke, vol. xlvii. p. 287) that these inte- 

 grating operators (Pot, Lap) are best expressed as inverse func- 

 tions of V. To see how utterly inadequate the Nabla would 

 have been to express the idea, we have only to imagine the 

 exclamation points which the members of the British Associa- 

 tion would have looked at each other if the distinguished 

 speaker had said : 



Helmholtz first solved the problem-Given the Nabla of the 

 velocity in any case of liquid motion, to find the velocity His 

 solution was that the velocity was the Nabla of the inverse 

 square of Nabla of the Nabla of the velocity. Or, that the 

 velocity was the inverse Nabla of the Nabla of the velocity. 



Or, if' the problem and solution had been written thus- 

 Required a in terms of Vw when Sva> = o. 

 Solution : 



NO. 1242, 



VOL. 48] 



Toi. 



My critic has himself given more than one example of unfit- 

 ness of the inverse Nabla for the exact expression of thought. 

 For example, when he tays that I have taken "eight distinct 

 steps to prove two equations, which are special cases of 



I do not quite know what he means. If he means that I have 

 taken eight steps to prove Poissoii's Equa ion (which certainly 

 is not expre-sed by the equation cited, ahhough it may perhaps 

 be associated with it in some minds), I will only say that my 

 proof is not very long, especially as I have aimed at greater 

 rigour than is usually thought necessary. I cannot, however, 

 compare my demonstration with that of quaternionic writers, as 

 I have not been able (doubtless on account of insufficient search) 

 to find any such. 



To show how little foundation there is for the charge that the 

 deficiencies of my system require to be pieced out by these 

 integral operators, I need only say that if I wished to economise 

 operators I might give up New, Lap, and Max, writing for them 

 y Pot, V X Pot, and v. Pot, and if I wished further to economise 

 in what costs so little, I could give up the potential also by using 

 the notation (v.v)"' or V"-. That is, I could have used this 

 notation without greater sacrifice of precision than quaternionic 

 writers seem to be willing to make. I much prefer, however, 

 to avoid these inverse operators as essentially indefinite. 



Nevertheless — although my cri ic has greatly obscured the 

 subject by ridiculinf; operatois, which I beg leave to maintain 

 are not worthy of ridicule, and by thoughilessly asserting that 

 it was nece-sary for me to use them, whereas they are only 

 necessaiy for me in thestnse in which something of the kind is 

 necessary for the qualernionist also, if he would use a notation 

 irreproachable on the scoie of exactness — I desireto be perfectly 

 candid. I do not wish to deny that the relations connected with 

 these notations appear a little more simple in the quaternionic 

 form. I bad, indeed, this subject principally in mind when I 

 said two years ago in Nature (vol. xliii. p. 512) : " There are 

 a few formulse in which there is a trifling gain in compactness 

 in the use of the quaternion.'' Let us see exactly how much 

 this advantage amounts to. 



There is nothing which the most rigid quaternionist need 

 object to in the notation for the potential, or indeed for the 

 Newtonian. These represent respectively the operations by 

 which the potential or the force of gravitation is calculated 

 from the density of matter. A quaternionist would, however, 

 apply the operator New not only to a scrlar, as I have done, 

 but to a vector also. The vector part of New ai (construed in 

 the quaternionic sense) would be exactly what I have repre- 

 sented by Lap w, and the scalar part, taken negatively, would 

 be exactly what I have represented by Max a. The quater- 

 nionist has here a slight economy in notations, which is of less 

 importance, since all the operators— New, Lap, Max — may be 

 expressed without ambiguity in terms of the potential, which is 

 therefore the only one necessary for the exact expression of 

 thought. 



But what are the formula; which it is necessary for one to 

 remember who uses my notations? Evidently only those which 

 contain the operator Pot. For all the others are derived from 

 these by the simple substitutions 



New = V Pot, 

 Lap = V X Pot, 

 Max = V. Pot. 



Whether one is quaternionist or not, one must remember 

 Poisson's Equation, which [ write 



V. V Pot a? = -45rw, 

 and in quaternionic might be written 



V^ Pot a — 4Tra). 

 If CO is a vector, in using my equations one has also to remem- 

 ber the general formulse, 



V. Vw = VV.« - V X V X w 

 which as applied to the present case may be united with the 

 preceding in the three-niembered equation, 



V. V Pot » = VV . Pot « - V X V X Pot 0) - - 4iriii. 

 This single equation is absolutely all that there is to burden 

 the memory of the student, except that the symbols of differen- 

 tiation (V, V X , V.) may be placed indifferently before or after 

 the symbol for the potential, and that if we choose we may 

 substitute as above New for v Pot, &c. Of course this gives a 

 good many equations, which on account of the impoitance of 



