246 



NA TURE 



[January i i, 1894 



most felt, corresponds with a series of outlying masses of car- 

 boniferous limestone, which are separate from the main mass of 

 carboniferous limestone of the Mendip anticlinal. Whether 

 these are parts of another anticlinal, or owe their position to 

 faulting, I do not know. Westward of Wells these outliers 

 form little knolls, as at Draycott and Westbury. Further east, 

 in the area between Wells, Shepton Mallet, and West Compton, 

 they form a group of prominent hills, whose valleys are occupied 

 by later formations. If such outliers exist east of Shepton 

 Mallet, they are deeply hidden by the oolitic strata. 



Evercreech and West Pennard lie off this carboniferous lime- 

 stone, but it extends beneath the valley in which these villages 

 lip. Priddy is on the main aniiclinal of the Priddy Mendip 

 hills, 3nd Chewton is separated from all the foregoing by the 

 exposure of Old Red Sandstone. It would be interesting to know 

 how far the Old Red Sandstone shared the movements ; but in- 

 formation is likely to be scanty, as the sandstone forms a bleak 

 and sparsely inhabited region. F. J. Allex. 



Mason College, Birmingham, January 6. 



On the other hand, when ^ is rotational, let its conjugate be 

 <^', then instead of (i) we have 



Quaternionic Innovations. 



That Prof. Tait should not be able to do justice to those 

 who prefer to treat vectors as vectors, and quaternions as 

 quaternions, instead of commingling their diverse natures, with 

 the result, in the latter case, of confusion of physical ideas 

 (and geometrical also, for of course geometry is itself ultimately 

 a physical science, having an experiential foundation), is 

 naturally to be expected. He does not know their ways, 

 either of thinking or of working, as is abundantly evident in all 

 that he has written adversely to Prof. Willard Gibbs and others. 

 It is, however, a little strange, in view of Prof. Tait's often 

 expressed conservatism regarding Quaternionics, that he should 

 tolerate auy innovations therein, such as Mr. MacAulay has 

 introduced. The latter may perhaps take this as a compliment 

 to his analytical powers, which compel the formers admiration, 

 and toleration of his departures from quaternionic usage. For 

 myself, I welcome any quaternionic innovations that may 

 (ultimately) tend in the direction of the standpoint assumed by 

 Prof. Gibbs and others, and foresaw some two years since 

 (when a very bulky manuscript came to me for my opinion) 

 that there would be some quaternionic upstirring. 



Prof. Gibbs has already pointed out how the development of 

 Quaternionics has involved first the elimination of the imaginary, 

 and next the gradual elimination of the quaternion ! Now 

 there is a capital illustration of this innate tendency in Prof. 

 Tait's review (Nature, December 28, 1893), where, on p. 

 194, he explains by an example the meaning of a startling inno- 

 vation of Mr. MacAulay's. Put it, however, in vectorial form, 

 and let us see what it comes to then. Take the case of a stress 

 and the force to correspond (which is a little easier than Prof. 

 Tait's example, though not essentially different). Let ^ be a 

 stress operator (pure, for simplicity), so that (^N, or N<|), is the 

 stress per unit area on the N plane, N being any unit vector. 

 Now we know, by consideration of the stresses acting upon the 

 faces of a unit cube, that the N component of the force F per 

 unit volume is the divergence of the stress vector for the N 

 planes. That is, 



FN = V<?>N, (l) 



for any direction of N. I employ my usual notation for the 

 benefit of readers (now becoming numerous) who, though they 

 cannot follow the obscure quaternionic processes, can under- 

 stand the plainer ones of pure vector algebra. Now, may we 

 remove the vector N (which is any one of an infinite number 

 of vectors) and write 



F = V^ or —<pv (2) 



simply, as the complete expression for the force ? Certainly we 

 may. For, in full, we have 



V = iVa+jVo4-kV3, (3) 



<^=:^i.i-i-^i.j + ^3.1t or =i.^j-j-j.02 ;-lt.<|)3, (4) 



where Vi, &c. are the scalar components of the vector v (not a 

 quaternion, of course) and ^j, &c. are the vector stresses on the 

 planes of i, &c., so that <p^ — <pS., &c. Direct multiplication 

 gives at once 



V«?) = Vl<?)j+Vn(?)o + V3<?)3, (5) 



which is F. We may also write it <pv, because ^ is pure. 



NO. 1263. VOL. 49] 



and therefore 



FN = V<|>'N, 



(6) 



(7) 



Here if ^ is given by the first expansion in (4), ^' is given by 

 the second. 



Now there are several things that deserve to be pointed out 

 about the above, which should be compared with Prof. Tait on 

 p. 194. First, that the result F = 4>v, irrespective of pureness, 

 or F = v^ also when the stress is pure, when got quaternioni- 

 cally seems to be a great novelty to Prof. Tait, and to give him. 

 a " severe wrench," involving a " dislocation " and a " startling 

 innovation." Perhaps, however, it is only Mr. Macaulay's 

 peculiar way of arriving at the result, that Prof. Tait is alluding 

 to. Moreover, secondly, in the vector algebra of Willard 

 Gibbs and others the use of equation (2) or of (7) to express the 

 force complete, by removal of the intermediate vector N, is 

 neither new, nor does it involve any straining of the intellect, 

 for it is actually apart of the system itself, done naturally and in 

 harmony with Cartesian mathematics. See Gibbs's " Elements 

 of Vector Analysis" (1881-4) for the direct product of vand^. 

 (Also for the skew product, a more advanced idea ; it, too, is 

 a physically useful result. ) Thirdly, note how very differently 

 the same thing presents itself to Prof. Tait according as it is 

 clothed in his favourite quaternionic garb or in vectorial vest- 

 ments. In the latter case it is either unnoticed or is con- 

 temptible ; in the former, it may be a novel and valuable 

 improvement. 



I do not think that Prof. Tait does justice to Mr. MacAulay 

 in making so much of a trifle such as passes unnoticed or un- 

 appreciated in the previous work of others. There is, I know, 

 much more in Mr. MacAulay's mathematics than Prof. Tait has 

 yet fathomed. For my own part, I like to translate it into 

 vectors, not merely because it is then in a form I am used to, 

 and is plainer, but also because the true inwardness of these 

 processes involving linear operators is properly exhibited by the 

 dyadical way of viewing them in conjunction with vectors, 

 without the forced and unnatural amalgamation with quaternions, 

 and the attendant obscurities. This seems to me to be par- 

 ticularly true in physical applications. I should not be writing 

 this note were it not for the misconceptions that Prof. Tait 

 indulges in about what he does not know, viz. vector algebra 

 apart from quaternions. At the same time, to avoid possible 

 misunderstanding, I disclaim any hostility to Mr. MacAulay's 

 quaternionic innovations, although I must agree with Prof Tait 

 as to the "singular uncouthness" of some of his expressions 

 in their present form. I hope he may be able to see his way to do 

 his work vectorially. It will be more amenable to innovations, 

 I think, without mental wrenches. At any rate he is a reformer, 

 and not afraid to innovate when he thinks fit. 



Oliver Heaviside. 



Paignton, Devon, December 30, 1893. 



The Second Law of Thermodynamics. 



I APOLOGISE to Mr. Bryan for unintentionally reading into 

 the Report, Article 17, what he did not intend to be there. I 

 understand now that according to his view conservative systems 

 are not alone to be included in the Clausian proof. 



My point, however, is (or was) that they ought to be ex- 

 cluded, at all events when there is only one controllable co- 

 ordinate V, because (l) in conservative systems the virial 

 equation gives a relation between T and v, so that only one of 

 them is independent. That, I submit, is true in fact. And (2) the 

 second law, I said, requires two independent variables. That, 

 however, is a question of definition, and if Mr. Bryan were to 

 take the equation 



r9Q _ „ 



for a complete cycle, whatever be the nature of the system, as 

 a definition of the second law, I see no valid objection to that 

 definition. 



I admit, and did admit, that for a conservative system, moving 

 in a complete cycle, 



■aQ_ 



T 



\' 



o, 



