392 



NATURE 



[December io, 1914 



others decorated for bravery in the field — the Soci^t^ 

 d'Anthropologie still continues its meetings. I was 

 able to inform Prof. Manouvrier that the sister society 

 in London, in common with other English scientific 

 societies, also continued its meetings, that many of 

 its members — including the honorary secretary and 

 assistant-secretary, had volunteered for active service, 

 and that a member of the council (Major S. L. Cum- 

 mins) had been decorated with the Legion of Honour. 



" Pour moi," writes Prof. Manouvrier, in conclud- 

 ing his letter, "je ne puis plus qu'un v6t^ran — un 

 ancien volontaire de 1870. Je fus ainsi un tres obscur 

 compagnon d'armes de notre aim6 Lord Kitchener." 



Royal College of Surgeons, Arthur Keith. 



Lincoln's Inn Fields, W.C. 



A Central Thought of Vector Analysis as used 

 in Physics. 



Vector analysis is now recognised fully as furnish- 

 ing standard modes of expression for many ranges 

 of mathematical and physical thought ; and mean- 

 while, the atmosphere of the physical thought especi- 

 ally is continually reacting upon the newer analytic 

 method, and adapting it with preciser discrimination 

 to the needs of actual use. Thus the dichotomous 

 arrangement of quantities as scalar or vector has 

 been supplemented by the tensor-triplet as a tertium 

 quid, vectors are habitually grouped as free or local- 

 ised, pseudo-scalars have been set off from pure 

 scalars. Something has been done, too, through the 

 influence of ascertained physical connections, toward 

 unifying discrete formal results by appreciating real 

 underlying articulation. Such advances as are made 

 deserve immediate incorporation in the accepted text, 

 as likely to render the methods more widely avail- 

 able ; and the serving of that end has suggested 

 presenting the notes that follow. 



Most of the orthodox lines of development leave 

 scalar product and vector product isolated from one 

 another, each sharply differentiated by its own special 

 definition. And an impression is then ordinarily felt 

 that the novel characteristics of vector analysis in 

 cutting through the rules of algebra centre upon the 

 vector product. What there may be of exaggeration 

 in this impression can be corrected by very simple 

 and accessible considerations. First we may observe 

 that an axial vector and a vector product are essen- 

 tially two aspects of the same conception, the former 

 being potentially resolvable into two factors determining 

 an oriented area that is at the same time the geo- 

 metrical basis of the vector product. .But the pro- 

 jection of areas and their graphic representation by 

 their measured normals are familiar elementary 

 notions ; while the assignment of " circulation " 

 round the perimeter to an area, being necessary in 

 order to complete the plan of conventional representa- 

 tion, must be expected to appear also in the equiva- 

 lent vector product. In fact, the rule for sequence 

 of the factors there and the change of sign upon 

 reversal of that sequence are only assignment of cir- 

 culation scarcely disguised. To establish the con- 

 nection is to restore the emphasis of Grassmaun's 

 thought where it may have been neglected. 



At another point current procedure is apt to dis- 

 locate the vector product from its natural or original 

 setting, and encourage a misleading effect of anti- 

 thesis where there is in reality more nearly a com- 

 plementary relation. When a scalar product and its 

 corresponding vector product are introduced separately 

 and held apart by exclusive stress upon their diver- 

 gence of type, an instructive link uniting them is lost 

 sight of or ignored. At their source they are linked 

 in that completer combination of two vectors accord- 

 ing to the fundamental idea of multiplication of 

 which they are partial aspects. Again it is Grass- 

 NO. 2354, VOL. 94] 



maun who more than hints at this very point of view, 

 presented also in a somewhat different dress by the 

 quaternion and the dyad. For purposes of elementary 

 introduction the relation in question comes, perhaps, 

 most clearly into prominence if the vector factors to 

 be combined are from the outset of this particular 

 development written each as a " semi-cartesian " 

 trinomial, yielding thus nine terms for the presenta- 

 tion of the completed combination. On inspection 

 three of these terms are obviously scalar; they con- 

 stitute the " scalar product " (or expressed more fully 

 the " scalar side of the product "). The remaining 

 six terms are just as obviously to be accepted as 

 vectors in their nature; together they constitute the 

 "vector product" (or the complementary "vector 

 side of the product "). These mutual complements 

 make up the complete expression into which they 

 enter, though they cannot coalesce, very much as a 

 complex quantity contains a real term and a detach- 

 able imaginary term. 



The value of the more inclusive statement is to be 

 rated still higher in its justifiable extension to the 

 combination of Hamilton's operator with a field- 

 vector, which is .also expressible formally as the ex- 

 panded product of two trinomials, the nine terms of 

 which, here too distinguishable as scalar and vector to 

 unstrained interpretation, are presently grouped under 

 the headings "divergence" and "curl." For the 

 complete combination that might be written fvv) 

 is intended to place in our hands by its results a 

 means of specifying with all reasonably attainable 

 detail the local peculiarities of distribution for the 

 field-vector. And here the usual partition of the 

 specification into apparently quite disconnected 

 divergence and curl seems plainly artificial at first 

 sight, and becomes yet more so as we trace out the 

 intimately complementary character of the two items 

 in the description. It will always be helpful to eman- 

 cipate ourselves from the formal constraints of our 

 algebra, which here, for example, in a degree forbid 

 the association of scalar and vector parts, and aim 

 on the contrary, through the choice of orthogonal 

 components, at a segregation at once complete and 

 reduced to simplest terms. We remember with profit 

 that one main office of the vector analysis is to do 

 away with those mechanical features of algebra 

 which contribute nothing or little to the progress of 

 thought. Frederick Slate. 



University of California. 



Science and National Needs. 



In a recent issue of Nature (October 29, p. 222) 

 the remark occurs : — 



" In this hour of national emergency there is no 

 time to be lost. We cannot all be soldiers, but we 

 can all help, we men of science, in securing victory 

 for the allied armies. Every day lost means the 

 destruction of a number of our fellow-countrymen and 

 of our allies, and the sooner we co-operate for the 

 good of the nation the sooner will the war be over." 



Acting upon the suggestion, and bearing in mind 

 the manner in which the dew-ponds on Salisbury Plain 

 suffered during the drought of 1911, I immediately 

 offered my services to the War Office, to advise as to 

 the digging of dew-ponds and the choice of sites for 

 them. 



I find that there is not now so great a necessity for 

 thom there as formerly, as water has been laid on to 

 many of the camps, but should the necessity arise 

 anywhere that camps are placed, the War Office has 

 my offer before them. 



I write this in order to encourage others to do what 

 in them lies. Edward A. Martin. 



285 Holmesdale Road, South Norwood, S.E. 



