500 



NATURE 



[October 12, 1911 



.Mr. H. Bateman then contributed a note on the trans- 

 formation of an electromagnetic field into itself. It is 

 hoped that a discussion of the infinitesimal transformations 

 will lead to equations of motion which will complete the 

 electromagnetic scheme. Following the work of Mr. 

 Hargreaves on the effect of an infinitesimal transformation 

 on certain integral forms, it is assumed that these integral 

 forms are invariants for the infinitesimal transformation. 

 The analysis then indicates that two forms are exact 

 differentials. Some types of infinitesimal transformations 

 satisfying the conditions were obtained for particular 

 electromagnetic fields, and the transformations were inter- 

 preted geometrically. 



An account of the report of the committee for the further 

 tabulation of Bessel and other functions was given by Mr- 

 Nicholson. In the report tables are given (calculated bv 

 Mr. J. R. Airey) of the Neumann functions G„(x) and 

 Y„(x) to seven decimal places for n = o and n=l, and for 

 values of the argument from 01 to 160 by intervals of 

 o-i. During the course of the vear Sir G. Greenhill has 

 brought forward a scheme for the rearrangement of the 

 elliptic functions tables on a new basis. This scheme has 

 now received the approval of the association, and a grant 

 has been made towards the expenses of computation. 



In this department Mr. H. Bateman contributed a 

 paper on a geometrical theorem connected with six lines 

 in space. PP', QQ', RR' are three pairs of lines in space. 

 LL' are the common transversals of QQ', RR' ; 

 MM' the common transversals of RR', PP' ; and NN' the 

 common transversals of PP', QQ'. If QQ'RR' belong to a 

 regulus, then MM'NN' also belong to a regulus. 



Mr. H. Hilton read a paper on the canonical form of 

 an orthogonal substitution. After pointing out a short 

 method of reducing a real orthogonal substitution to a 

 canonical form, he discussed the analogous problem for an 

 orthogonal substitution with complex coefficients. A 

 canonical form was obtained in which the linear equations 

 of the substitution were arranged in blocks, some of which 

 contained two and others only one member. The leading 

 coefficient occurring in one type of block was the reciprocal 

 of the corresponding coefficient occurring in an associated 

 block. 



Prof. J. C. Fields read a paper on proof of certain 

 theorems relating to adjoint orders of coincidence, viz. : — 

 (i) In the reduced form of a rational function of (c, it), 

 which is adjoint for the value z = a (or z=<*-), the co- 

 ffin ient of » " ' is integral with regard to the elements 



z — a (or ). (2) If a rational function is adjoint for the 

 value e= c«, the degree <•! iis reduced form is <N — i. 

 (3) The reduced form of a rational function adjoint for the 

 value E = o is integral with regard to the element z — a if 

 the equation f(z, «) = («— PJ . . . . («-P„) = o (where 



P, • . . P„ are power-series in an element z — a (or-) with 



exponents integral or fractional) is integral with regard to 

 this element. 



The Principle of Relativity. 

 The proceedings on Friday, September i, opened with 

 a discussion on the principle of relativity, led by Mr. E. 

 Cunningham, who pointed out that the scope of the hypo- 

 thesis of relativity is exactly coincident in extent with its 

 scope in Newtonian dynamics. The acceleration of a 

 point is not physically indeterminate as its velocity is. 

 The theory of relativity is, for example, quite consistent 

 with the magnetic effects apparent to terrestrial observers 

 being explained as arising from the rotation of the earth 

 with a nearly stationary distribution of charge. Within 

 the limits indicated above and within the realm of pheno- 

 mena in which the sole determinative laws are those of 

 the electron theory, the hypothesis becomes a mathematic- 

 ally demonstrable fact in the sense that it is not po ibli 

 to choose a unique frame of reference for which alone the 

 laws will hold good. Mr. Cunningham then sketched the 

 transformations connecting the measurements made accord- 

 ing to two frames of reference, each of which is equally 

 adequate to express known phenomena, and explained what 

 deductions can be drawn. 



In the subsequent discussion Dr. W. F. G. Swann pointed 

 out that the compliance of a system with the electro- 

 magnetic scheme is by no means a suffii ient criterion for 



NO. 2l8q, VOL - 87] 



the possibility of its existence ; for instance, a system of 

 two singularities, moving along at a constant distance 

 apart, with the field at each point the sum of the ordinary 

 fields due to the separate motions, is a system in accord- 

 ance with the scheme, but it is one impossible of existence 

 in practice. The explanation of the impossibility of the 

 existence of such systems is to be found in the fact that 

 they could never evolve out of any actually existing system. 

 In fact, if we take the electromagnetic scheme as complete, 

 those uniformities in nature which we call " laws " are 

 to be looked upon as due in part to the compliance of the 

 universe with the scheme and in part to the individuality 

 of the initial system started. A complete knowledge of the 

 field at every point in space, both inside and outside the 

 molecules, would, in conjunction with the electromagnetic 

 scheme, theoretically give us all that we should need to 

 ascertain both the past and subsequent history of the 

 universe. Since we cannot know the complete field at 

 some instant, we are driven to make up for our incom- 

 plete knowledge by formulating certain " subsidiary laws," 

 such as laws involving the conception of forces betv 

 the singularities, electrical surface conditions, gravitation, 

 &c, the function of these laws being to restrict those 

 systems which satisfy the scheme, and are also to be con- 

 sidered as possible of existence, to those which can spon- 

 taneously evolve out of the actually existing universe. 



Mr. H. Bateman emphasised the mathematical interest 

 attached to the principle, because it unites several bram 

 of mathematics, such as geometry, partial differential 

 equations, generalised vector analysis, continuous groups of 

 transformations, differential and integral invariants, &c. 

 He pointed out that there are two different types of trans- 

 formations which can be used to transform one mathe- 

 matical specification of an electromagnetic field into 

 another. Those transformations which depend upon the 

 electromagnetic fields which they can be used to trans, 

 form form a much wider class than the spherical wave 

 transformations, which can be applied to any electro- 

 magnetic field. The new transformations provide us with 

 some very interesting analogues of mental phenomena. 



Prof. F. Zeeman stated that the value of Fresnel's 

 coefficient is easily deduced by means of the principle of 

 relativity provided that no account is taken of dispersion. 

 In that case the results of Fizeau's and Michelson and 

 Morley's experiments on the propagation of light in flow- 

 ing water agree with theory. The agreement is not so 

 good if a dispersion term formally calculated by Lorentz 

 be introduced. He asked whether the correction term 

 would be the same if the principle of relativity were applied 

 from the beginning to a dispersive medium. Prof. G. N. 

 Lewis outlined some of the views which he has recently 

 expressed in full in The Philosophical Magazine. Prof. 

 A. E. H. Love warned one to be on the alert, because in 

 the factor K = (i — /3 2 )-£ we may be dealing only with a 

 first approximation. Secondly, he suggested that it was 

 conceivable that terrestrial magnetism might be only an 

 apparent phenomenon due to the rotation of the earth. He 

 had, however, tested whether by taking rotatory axes one 

 could obtain effects of the magnitude of terrestrial 

 magnetism : but they turn out to be very small, and it 

 seems hopeless to think of magnetism as due solely to 

 the rotation. 



Dr. C. V. Burton, after expressing his satisfaction that 

 none had confessed a disbelief in the aether, urged the 

 importance of a search for residual phenomena not falling 

 within the electromagnetic scheme. Conceivably gravita- 

 tion is such a phenomenon. There is, further, a question 

 as to whether neighbouring electrically neutral masses exert 

 forces upon one another in virtue of their motion through 

 the aether. Such forces would be non-electromagnetic: 

 experiments designed to detect them are in progress, bul 

 have so far given a null result. The entire absence of 

 such forces would imply that matter takes up no room 

 (positive or negative) in the aether. 



So much time had been taken up by the discussion that 

 none was left to Mr. Cunningham to reply adequately to 

 the numerous points raised. His scholarly paper intro- 

 ducing the discussion has been ordered to he printed 

 in extenso ; we look forward to perusing it in its extended 

 form, and no doubt the answers to many of the questions 

 I will appear therein. 



