90 



NATURE 



[^ 



\_May 23, 1878 



will cause not only much fear and trembling among the 

 crammers but also perhaps very legitimate trepidation 

 among the august body of Mathematical Moderators and 

 Examiners. For, although (so far as we have seen) the 

 word quaternion is not once mentioned in the book, the 

 analysis is in great part purely quaternionic. And it is 

 not easy to see what arguments could now be brought 

 forward to justify the rejection of examination-answers 

 given in the language of quaternions — especially since in 

 Cambridge (which may claim to lay down the law on such 

 matters) Trilinear Coordinates, Determinants, and other 

 similar methods were long allowed to pass unchallenged 

 before they obtained formal recognition from the Board 

 of Mathematical Studies. 



Every one who has even a slight knowledge of qua- 

 ternions must allow their wonderful special fitness for 

 application to Mathematical Physics (unfortunately we 

 cannot yet say Mathematical Physic /) : but there is a long 

 step from iuch semi-tacit admissions to the full triumph 

 of public recognition in Text-Books. Perhaps the first 

 attempt to attain this step (in a book not ostensibly qua- 

 ternionic) was made by Clerk-Maxwell. In his great 

 work on Electricity all the more important Electrodynamic 

 expressions are given in their simple quaternion form — • 

 though the quaternion analysis itself is not employed : — 

 and in his little tract on Matter and Motion (Nature, 

 vol. xvi. p. 119) the laws of composition of Vectors are 

 employed throughout. Prof. Clifford carries the good 

 work a great deal farther, and [if for this reason alone] 

 we hope his book will be widely welcomed. 



To show the general reader how much is gained by 

 employing the calculus of Hamilton we may take a couple 

 of very simple instances, selecting them not because they 

 are specially favorable to quaternions but because they 

 are familiar in their Cartesian form to most students. 

 Every one who has read Dynamics of a Particle knows 

 the equations of non-acceleration of moment of momentum 

 of a particle, under the action of a single centre of force, 

 in the form 



xy — yx = o 



y'z — zy = o 



zx — x'z = o 



with their first integrals, which express the facts that the 

 orbit is in a plane passing through the centre, and that 

 the radius-vector describes equal areas in equal times. 

 But how vastly simpler as well as more intelligible is it 

 not to have these three equations written as one in the 

 form 



Vpp = o 



and]^the three first integrals above referred to as the 

 immediate deduction from this in the form 



Vpp = a. 

 Take again Gauss's expression for the work done in 

 carrying a unit magnetic pole round any closed curve 

 under the action of a unit current in any other closed 

 circuit. As originally given, it was 



'(^i - x) !,iydz^ - d2dy'')+(y^ -ytdzdx-^ -djgdz^)+(i!^ - 2) (d.rdy - dydjc^) 



// 



With the aid of the quaternion symbols this unwieldy 

 expression t:ikes the compact form 

 ' S.pdpii'p 



If 



7>» 



The meanings of the two expressions are identical, and 

 the comparative simplicity of the second is due solely to 

 the fact that it takes space of three dimensions as it finds 

 it ; and does not introduce the cumbrous artificiality of 

 the Cartesian coordinates in questions such as this 

 where we can do much better without them. 



In most cases at all analogous to those we have just 

 brought forward. Prof. Clifford avails himself fully of the 

 simplification afforded by quaternions. It is to be re- 

 gretted, therefore, that in somewhat higher cases, where 

 even greater simplification is attainable by the help of 

 quaternions, he has reproduced the old and cumbrous 

 notations. Having gone so far, why not adopt the 

 whole ? 



Perhaps the most valuable (so far at least as physics is 

 concerned) of all the quaternion novelties of notation is 

 the symbol 



dx dy d 2 



whose square is the negative of Laplace's operator : i.e. 



A glance at it is sufficient to show of what extraordinary 

 value it cannot fail to be in the theories of Heat, Electri- 

 city, and Fluid Motion. Yet, though Prof. Clifford 

 discusses Vortex-Motion, the Equation of Continuity, 

 &c. we have not observed in his book a single V- There 

 seems to be a strange want of consistency here, incoming 

 back to such "beggarly elements" as 



instead of 



- Sva, 



especially when, throughout the investigation, we have 

 <r used for 



ui + vj + wk ; 



and when, in dealing with strains, the Linear and Vector 

 Function is quite freely used. Again, for the vector axis 

 of instantaneous rotation of the element at x,y, z (p), 

 when the displacement at that point is or = iu -^-jv + 

 k7v, we have the cumbersome form 



i {(5y TV — Sz^v) i + % u — ix iv)j + (hxV — 5y u) k\ 

 instead of the vastly simpler and more expressive 

 \ Vv<r. 



It may be, however, that this apparent inconsistency is 

 in reality dictated by skill and prudence. The suspicious 

 reader, already put on his guard by Clerk- Maxwell's first 

 cautious introduction of the evil thing, has to be treated 

 with anxious care and nicety of handling : — lest he should 

 refuse altogether to bite again. If he rise to the present 

 cast we shall probably find that Prof. Clifford has v, in 

 the form as it were of a gaff, ready to fix him irrevo- 

 cably. That he will profit by the process, in the long run, 

 admits of no doubt :— so the sooner he is operated on 

 successfully the better. What is now urgently wanted, 

 for the progress of some of the most important branches 

 of mathematical physics, is a "coming" race of intelli- 

 gent students brought up, as it were, at the feet of 

 Hamilton ; and with as little as may be of their freshness 

 wasted on the artificialities of x,y,z. Till this is pro- 

 cured, quaternions cannot have fair play. Nut-cracking, 



