sS 



NA rURE 



[February 9, 1899 



SOCIETIES AND ACADEMIES 

 London. 



Royal Society, January 19.— "On the Vibrations in the 

 Kield round a Theoretical Hen/ian Oscillator." By Karl 

 Pearson, K.R.S., and .\licc Lee, li.A. 



The object of this paper is to investigate the types of wave 

 motion in the neighbourhood of a ///(W/r/rVij/ Hertzian oscillator. 

 By a theoretical Hertzian oscillator the writers understand a 

 Maxwellian "double point " of initial maximum moment + E/. 

 But as the actual oscillator has been shown by Bjerknes and 

 others to give a damped wave train, they take the maximum 

 moment to run down with the time, and to oscillate between the 

 limits + E/t~''i'. This gives a wave train corresponding to that 

 observed by Bjerknes and represented at a given distance by 



Ce-i'i' sin (/./ + y). 

 The investigation for a "double point" with a steady wave 

 train was originally made by Hertz himself, and has found its 

 way into most of the current textbooks of electro-magnetism. 

 The theory there given, is insufticient for two reasons, both of 

 which were recognised by Hertz himself, namely, because (i. ) 

 the actual oscillator has sensible extension, and (ii,) the wave 

 train it gives forth is not steady. 



The present jiaper only attempts to remove the latter objection 

 to Hertz's original theory ; like that theory it becomes less 

 accurate as we approach nearer to an actual oscillator. The 

 chief divergences between the present and Hertz's original 

 theorj' actually fall in that portion of the field wherein his chief 

 interference experiments were made. 



The writers investigate the general theory of a double point 

 with damped intensity, and replace the well-known Hertzian 

 diagrams of the field by a more complete series of 56, representing 

 the field for seven complete oscillations, and showing how the 

 field for some twelve metres round the oscillator chosen, gradu- 

 ally falls to nearly j'b of its maximum initial strength. These 

 diagrams are entirely due to Miss Alice Lee, and involved a 

 large expenditure of labour and time, which would, perhaps, 

 not have been justified were any other graphic representation of 

 a damped wave motion available. 



The writers next deal with the type of waves propagated, 

 tlieir velocities and their phases. 'The following general con- 

 clusions are reached : — 



(i.) Three waves of electro-magnetic force may be considered 

 as sent out from the oscillator. These are : — 



(a) A wave of purely transverse electric force. 



(b) A wave of electric force parallel to the axis, briefly termed 

 the wave of axial electric force. 



((■) A wave of magnetic force. 



The waves of axial electric and of magnetic force move out- 

 wards with the same velocity, which is, however, a function of 

 the distance from the centre of the oscillator. The intensity of 

 both forces for points on the same sphere varies as the cosine of 

 the latitude, the polar axis being the axis of the oscillator. 



The wave of transverse electric force is propagated with the 

 same velocity at all equal distances from the centre of the oscil- 

 l.ator, but this velocity difiers from that of the two previous waves ; 

 further, the amplitude is independent of the latitude, being con- 

 stant over any sphere. The velocity after the wave has reached 

 a certain distance from the double point is always greater than 

 that of the waves of magnetic and of axial electric force. Its ex- 

 cess over the velocity of light tends to become three times the 

 excess of the velocity of the magnetic wave over the velocity of 

 light ; Ixjth the excesses decreasing asymptotically. 



(ii.) The velocities of these waves undergo remarkable changes 

 in the neighljourhood of the oscillator, but these changes extend 

 to distances which are greater than those within which a great 

 proportion of Hertz's interference experiments were made. 



(iii.) The point of zero phase for both transverse and axial 

 electric waves does not coincide with the centre of the oscillator, 

 so that these waves appear to start from spheres of small but 

 finite radius round the oscillator. .\ fourth wave dealt with by 

 Hertz, namely, the wave of magntiic induction, does not, as he 

 supposes, start with zero phase from the origin, but with a finite 

 phase. The wave in the equatorial plane, largely relied upon 

 Dy Hertz for his interference experiments " of the first kind," 

 is a compound of the waves of transverse and axial electric 

 force, and has a much more complex series of velocity changes 

 than Hertz appears to have realised. 



NO. 1528, VOL. 59] 



(iv.) The existence of the two electric force waves and the 

 singular changes of the wave motion in the neighbourhood of 

 the oscillator very possibly throw light on the difficulties which 

 arise in Hertz's experiments. It would seem thai such experi- 

 ments should be made at distances greater than 6 to 7 (Kjzir) 

 from the centre of the oscillator, or, roughly, about a wave- 

 length from the oscillator. In Hertz's case this amounts to 

 about 10 metres — a distance at which Hertz rather terminated 

 than started his interference experiments. 



February 2. — "Sets of Operations in Relation to Groups 

 of Finite Order." By .\. N. Whitehead, .M.A. Communi- 

 cated by Prof. A. R. Forsyth, F.R.S. 



The present paper is concerned with the Theory of Groups of 

 Finite Orders. The more general object of the paper is to 

 place this theory in relation to a special algebra of the type con- 

 sidered in the general theory of Universal .Algebra. This 

 special algebra, which may be called the .\lgebra of Groups of 

 Finite Order, has many affinities to the .\lgebra of Symbolic 

 Logic ; and a comparison of it with this algebra is given in the 

 last section of this paper. 



The N symbols, or operations, are considered to be capable 

 of addition according to the law 



S-fS = S. 



This is the well-known law of addition in Symbolic Logic, 

 and the introduction of numerical symbols as factors is thereby 

 avoided. 



The sum of a selection of the N fundamental operations, such 

 as S^ -h Sj -H S, -f- S„ is called a set. If a set obeys certain 

 special conditions it is called a group. The sum of the whole 



number (N) of fundamental operations, namely, So + S,-f 



-^Ss_l, obeys these conditions. This sum is called the complete 

 group, and all other groups are its sub-groups. 



The first six sections of this paper are devoted to the detailed 

 establishment of this purely algebraic view of the subject. At 

 times the modification in treatment from that adopted in the 

 standard treatises on the subject, such as Burnside's " Theory of 

 Groups of Finite Orders," is slight. 



The more special object of this paper follows directly from 

 the changed point of view from which the Theory of Groups is 

 here regarded. The idea of the group is no longer so absorbing ; 

 the set takes its place as the fundamental general entity which 

 has to be investigated. Accordingly in this paper some of the 

 general properties of sets are investigated. A set of operations 

 has numerous groups associated with it, and these groups have 

 many relations with each other. The fundamental idea of this 

 part of the paper ((/ § 7) is the formation from a .set H of an 

 unending series of other sets, here called the successive powers 



of H, and in the notation of the algebra written H'-', H', 



This series is called the power sequence of II. Any group 

 which contains H also contains its power sequence. The 

 power sequence is proved to have a periodic property (cf. § 9) 

 which introduces a curious analogy to recurring decimals. This 

 periodicity is the foundation of the rest of the paper. It 

 governs the relations to each other of the various allied groups 

 and sets. The periodicity is expressed by an e(|uation of the 

 form 



H"+'">+v = H"+'', 



where m is called the period of H, and « the characteristic, 

 and s and </ are any integers including zero. The number 

 of theorems relating to in is very large. 



Linnean Society, January 19. — William Carruthers, 

 F.R.S., \ice- President, in the chair. — .Mr. H. W. .Monckton 

 exhibited specimens of Mya arenaria, Linn., from Norway. 

 Heand Mr. R. S. Herries (Sec. Geol. Soc.) had found a colony 

 of these molluscs living on a sand-flat at the head of the 

 Fja;rland Fjord, about eighty miles from the open sea and where 

 the water at the surface is fairlj- fresh. The great snowfield, the 

 Sostedal, approaches close to the north-west side of the (jord, 

 and at a level of only 3500 feet to 4000 feel above it, where 

 glaciers descend into the valleys at the head of the fjord to 

 within four miles of the mud-flat in question. The shells were 

 for the most part small and thin, and this might be due to the 

 freshness or to the coldness of the water, or both. — Dr. W. G. 

 Kidewood read a paper, entitled "Some Observations on the 

 Caudal Diplospondyly of Sharks," from which he concludetl 

 that the occurrence of twice as many vertcbr.e as inuscle- 

 segmenls is a secondary feature, but one of ancient d.ate ; and. 



