October 22, 1903] 



NATURE 



601 



itself kindly to the explanation of such phenomena 

 as the propagation of transverse waves. The medium 

 considered in the present mvestigation is assumed 

 [o consist of uniform spherical grains which are so 

 close together as to prevent diffusion, and when in 

 a state of " normal piling " the centre of each grain 

 i-. supposed to be equidistant from those of twelve 

 neighbouring grains, this being the distribution 

 corresponding to minimum volume, and the system 

 " constituting to a first approximation an elastic 

 medium with six axes of elasticity symmetrically 

 placed." [It may be pointed out before proceeding 

 further that there is more than one way of piling balls 

 so that each ball is in contact with twelve neighbour- 

 ing ones and the total volume is a minimum.] 



The grains are supposed to be capable of limited 

 relative motion, and local inequalities may exist due 

 to the presence or absence of a number of grains above 

 or below that necessary for normal piling. In such 

 cases singular surfaces are formed between the grains 

 in unstrained and those in strained piling. The author 

 finds that the local negative inequalities produced by 

 the absence of grains present the ordinary properties 

 of matter. They are free to move through the medium 

 without resistance, the grains streaming freely through 

 their singular surfaces, and they attract one another 

 according to the law of the inverse square. The 

 density of matter is thus negative, taking that of the 

 medium to be positive, and if the density of water be 

 taken as — i, the author finds that of the medium to 

 be 10'. The diameter of the grains in C.G.S. units 

 is 5-534 X 10- '% their mean path is 8.612 x 10-==*, their 

 mean relative velocity is 6.777 x 10. the mean pressure 

 is 1. 172 X 10", the rate of propagation of the transverse 

 wave is 3004 x lo*", and the rate of degradation of 

 the transverse wave is such that it would require 56 

 million years to reduce the total energy in the ratio of 

 I to e\ The absorption thus produced is of such a 

 magnitude as to account for the blackness of the sky 

 on a clear night compatibly with the absence of any 

 measurable absorption of light by the ether. On the 

 other hand, the absence of any' evidence of normal 

 waves until quite recently is accounted for by the con- 

 clusion that the rate of degradation of the normal 

 wave would reduce its energy to about one-eighth in 

 3.923XIO-' of a second, or'before it had traversed 

 2200 metres. In addition to positive and negative in- 

 equalities of which the latter correspond to matter, 

 the existence is assumed of " complex inequalities " 

 due to the displacement of grains from one position 

 to another, and a comparison of the attractions of 

 such inequalities with those due to the inequalities 

 representing matter is in complete accordance with 

 the known smallness of gravitative as compared with 

 electric action. 



The theory accounts for the refraction, dispersion, 

 polarisation by reflection, metallic reflection and 

 aberration of light. 



The analytical investigation is based on the general 

 equations of motion and conservation of any entity 

 (Section ii.), these equations being generalisations 

 of the well-known equations of continuity of hydro- 

 dvnamical and allied systems ; the formation of 

 the equations of motion in a purely mechanical 

 medium (Section iii.), the separation of the motion 

 into its components of " mean " and relative motion 

 (Sections iv.-vii.), the extension of the kinetic theory 

 to granular media (Sections viii.-x.), and an elaborate 

 analysis of the changes taking place in the angular 

 inequalities, the momentum and energy, the mean and 

 relative systems, and the mean inequalities and their 

 motions (Sections xi.-xiv.). It should be observed that 

 X\\'i present theory involves the assumption that posi- 

 tively electrified bodies do not repel each other, and 

 for this the author gives arguments in § 226. In 



NO. 1773, VOL. 68] 



the final section (xv.) the numerical values of the 

 quantities which define the condition of the granular 

 medium, as stated above, are deduced from the results 

 of physical experience. 



The mathematical reasoning is very difficult, in some 

 places almost impossible, to follow, owing to the large 

 number of doubtful points or inaccuracies in the equa- 

 tions. Even if the fundamental conclusions should 

 prove to be correct, there are many points in the argu- 

 ment which are at present obscure, and require to be 

 ckared up. To take a few examples, in equation (4), 

 p. 10, a new symbol r is introduced without any ex- 

 planation, and the dual use of 5 is very confusing. 

 Having used 5S to denote a volume element, and 

 8s a surface element on this page, the author suddenly 

 changes from 6S to 5s in the first of equations (20) on 

 p 16, and to ds in the second and third, although he 

 refers to equation (2) of p. 10, which involves 5S. On 

 p. 13 in equation (13), the differential is omitted after 

 the treble sign of integration; also in (16) one of 

 the expressions under the sign of summation is multi- 

 plied by the differential element dS, while the other 

 is not ; in the former equation the reader will naturally 

 supply the missing dx dy dz, but the meaning of the 

 latter equation is obscure. Again, turning to p. 105, 

 we find that § 116 refers to " The mean velocities of 

 pairs having relative velocities s/ 2W ^' and Vj'/V2," 

 while in § 120 we read, " Since the mean velocities of 

 pairs of grains having relative velocity V^V/ is 

 V//y2 . . . ." In § 117, "All directions of mean 

 velocity of a pair are equally probable whatever the 

 direction of the mean velocity." On p. 120, equation 

 (181), it is not easy to see how, if N be equal to the 

 number of grains in unit volume, the square root of 

 N should be equal to N dx dy dz multiplied by a certain 

 function of the coordinates, nor how by integrating 

 the equation with respect to y and z the square root 

 of N now becomes equal to N multiplied by another 

 function multiplied by the linear differential dx. In 

 ordinary circumstances there is no useful purpose 

 served in filling a review with a list of errata which 

 any reader could easily correct for himself. But the 

 present investigation would be difficult to follow even 

 under the most favourable conditions, and the presence 

 of so many formulae and statements which cannot 

 possibly be correct as they stand renders the task well 

 nigh hopeless. 



An objection of an entirely different character applies 

 to the sections in which Maxwell's law of distribution 

 of velocity components and partition of energy is ex- 

 tended to a medium of closely packed spheres such as 

 that considered by Prof. Reynolds. \ great 'deal has 

 been written as to the validity of Maxwell's law, and 

 of the fundamental assumptions involved in the proofs 

 of it. The general opinion on which all mathematical 

 physicists are pretty well agreed is that the law holds 

 good to a first approximation in gaseous media the 

 molecules of which are not too closely crowded 

 together; but one method of proof after another has on 

 closer examination been found to involve some assump- 

 tion or other which usually breaks down in the case 

 even of a dense gas. Moreover, Mr. Burbury has gone 

 so far as to establish a different formula for the law of 

 distribution in dense gases. To assume the law to 

 hold good in the extreme case of a medium the ultimate 

 particles of which are permanently interlocked must 

 be regarded, failing other evidence than that given by 

 Maxwell, as a very doubtful step. 



A number of interesting questions suggest them- 

 selves for the consideration of physicists, such as 

 the ultimate distribution of energy between the grains 

 and molecules, the determination of the temperature 

 of cosmic space as defined by the mean kinetic energy 

 of the grains, the influence of the absorption of the 

 medium, however small, on the progress of cosmic 



