NA TURE 



193 



QUATERNIONS AS AN INSTRUMENT IN 



PHYSICAL RESEARCH. 

 Utility of Quaternions in Physics. By A. McAulay, 



M.A. (London : Macmillan and Co., 1893.) 

 T UST as " one shove of the bayonet " was truly said to 

 J be more effective than any number of learned 

 discussions on the art of war : — this really practical 

 work, giving genuine quaternion solutions of new pro- 

 blems as well as largely extended developments of old 

 ones, is of incomparably greater interest and usefulness 

 than the recently renewed, but necessarily futile, attempts 

 to prove that a unit vector cannot possibly be a quad- 

 rantal versor : — nay, that a Calculus of Vectors must 

 limit itself to the beggarly elements of addition and sub- 

 traction, commonly called "composition." 



It is much to be regretted that Mr. McAulay has not 

 determined simply to let his Essay speak for itself. His 

 Preface, though extremely interesting as the perfervid 

 outburst of an enthusiast, assumes here and there a 

 character of undignified querulousness or of dark insinu- 

 ation, which is not calculated to win sympathy. It has 

 too much of the," Rends-toi, coquin " to make willing 

 converts ; and in some passages it runs a-muck at 

 Institutions, Customs and Dignities. Nothing seems 

 safe. It is a study in monochrome : — the lights dazzlingly 

 vivid, and the shades dark as Erebus ! We gladly pass 

 rom it to the main contents of the book. 

 There can be no doubt whatever of its value from the 

 scientific point of view. It is the work of a man of 

 genuine power and originality. Many parts of it are, no 

 doubt, laboured and somewhat heavy, others very crude ; 

 and in some places the obscurity is almost repulsive. 

 [Curiously, these obscurities occur chiefly where more 

 than usual pains have been taken to make things plain !] 

 But faults like these are well-nigh inevitable in a first 

 effort ; and they should, perhaps, be regarded as 

 enhancing by contrast the merits of the novel processes 

 and results to which they act as a foil. 



It is positively exhilarating to dip into the pages of a 

 book like this after toiling through the arid wastes pre- 

 sented to us as wholesome pasture in the writings of 

 Prof. Willard Gibbs, Dr. Oliver Heaviside, and others 

 of a similar complexion. Here, at last, we exclaim, is a 

 man who has caught the full spirit of the Quaternion 

 system : — " the real csstiis, the awen of the Welsh bards, 

 the divinus afflatus that transports the poet beyond the 

 limits of sublunary things"! No doubt, to a man like 

 this, the restrictions imposed, in view of the prospective 

 ordeal of the Senate-House, by the passionless worldly- 

 wisdom of a " Coach," must have been gall and bitter- 

 ness. Intuitively recognising its power, he snatches up 

 the magnificent weapon which Hamilton tenders to all, 

 and at once dashes off to the jungle on the quest of big 

 game. Others, more cautious or perhaps more captious, 

 meanwhile sit pondering gravely on the fancied imper- 

 fections of the arm; and endeavour to convince a 

 bewildered public (if they cannot convince themselves) 

 that, like the Highlander's musket, it requires to be 

 treated to a brand-new stock, lock, and barrel, of their 

 NO. 1 261, VOL. 49] 



own devising, before it can be safely regarded as fit for 

 service. " Non //zi' juventas orta parentibus . . ." What 

 could be looked for from the pupils of a School like 

 that? 



Mr. McAulay himself has introduced one or two rather 

 startling innovations. But, unlike the would-be patchers 

 or underpinners to whom we have referred, he retains 

 intact all the exquisitely-designed Hamiltonian ma- 

 chinery, while sedulously oiling it, and here and there 

 substituting a rolling for a sliding contact, or introducing 

 a lignum vitce bearing. To borrow an analogy from 

 current electricity, he endeavours to add facilities, while 

 his concurrents are busy adding resistances, sometimes 

 indeed breaking the circuit altogether ! 



Among the additions to which Mr. McAulay calls 

 attention, some are certainly not novel, they were per- 

 fectly well known to Hamilton himself. Thus the use of 

 suffixes, to show which factor of a product (say) is to be 

 acted on by an operator, is at least as old as Herschel's 

 Appendix to the translation of Lacroix : — and is an 

 essential part of the notation required for what is cor- 

 rectly called "Hamilton's Theorem." Mr. McAulay 

 refers to this as a process of his own, which was found 

 "necessary somewhat to expand the meaning'' of a 

 symbol. Another instance is the use of a vector, which 

 may have an infinite nu?nber of values, for the purpose of 

 condensing three independent scalar equations into one 

 common expression, &c. This is purely and entirely 

 Hamiltonian. 



The "startling innovations," however, as we called 

 them above, are unquestionably Mr. McAulay's own — 

 and he has certainly gone far to justify their introduc- 

 tion. He has employed the sure tests of ready applica- 

 bility and extreme utility, and these have been well 

 borne. Objections based upon mere unwontedriess or 

 even awkwardness of appearance must of course yield 

 when such important advantages as these (if they be 

 otherwise unattainable) are secured ; but it certainly 

 requires a considerable mental wrench to accustom our- 

 selves to the use of 



d 



Xi 



dx-j^ 



as an equivalent for the familiar expression 



dx. 



If this be conceded, however, it is virtually all that 

 Mr. McAulay demands of us, and we are free to adopt 

 his system. It is to be carefully observed that there is no 

 interference with ihe. principles of quaternions to which, as 

 was remarked above, Mr. McAulay strictly adheres. The 

 quantities and operators, to which the dislocation applies, 

 are all scalars, and the wrench referred to is therefore an 

 algebraic, not a specially quaternion, one. Its introduc- 

 tion is made necessary by the determination to adhere to 

 the non-commutative property of quaternion multiplica- 

 tion, while endeavouring to effect certain desirable trans- 

 formations. Mr. McAulay likens this dislocation of the 

 usual arrangement of operator and subject to the 

 occasional disarrangement of relative position of adjec- 

 tive and substantive in a Latin sentence : — the nexus 

 between them being the common case-ending, which is 

 the analogue of the common suffix. A single example, 



K 



