6io 



NATURE 



[October 22, 1903 



attach to multiplication. In Prof. Henrici's algebra the 

 products of two vectors a, /3 are : — (a/3) a non-directional or 

 "scalar," in magnitude equal to the product of one 

 vector into the component of the other along the first, and 

 [o3] a vector perpendicular to the plane drawn through 

 a and ;9, and in magnitude equal to the area of the 

 parallelogram of which a and /3 are concurrent sides. 

 This algebra is evidently identical with those of Heaviside 

 and Gibbs, and, like them, open to the objection that it 

 does not discriminate between " polar " vectors, e.g. 

 forces and " axial " vectors, e.^. couples. Its relation 

 to that of quaternions is expressed by the equation 

 oj3= — (o;8)-f[a/8], where o;3 is the quaternion product of 

 o «nd /3. If, now, m be a scalar function of the vector p of 

 a point P, and P be displaced through a distance dp, the 

 change du. in the value of u will be proportional to dp, and 

 may be denoted by dp . V", where vu is a vector such that 

 for a given magnitude of dp, du is a maximum when dp 

 is parallel to V". Hence the direction of vm is that of 

 the greatest rate of change of u, and its magnitude that 

 rate of change. Similarly for a vector function r; of p 

 dpi] = (dp'v.)v, and v follows quite generally the laws of 

 combination of vectors. Thus we have (v»?) the " diver- 

 gence " of 71 and [ vr;] the '' curl " of tj, with their numerous 

 applications. By the use of this operator v, theorems like 

 those of Green and Stokes can be proved in a generalised 

 form with great ease and elegance, and the equations for 

 the electromagnetic field follow in a couple of lines of 

 work. 



With so powerful a calculus as this at command. Prof. 

 Henrici considers it the height of folly, after using 

 vectorial methods in those elementary parts of physics which 

 deal with addition of forces or velocities, to drop them 

 for Cartesian coordinates and direction cosines at the next 

 step forward. He advocates the use of vectors throughout, 

 and, like Heaviside, would make trigonometry follow and 

 depend on vectors by the definitions x = r cos e, y = r sin 0. 

 Vectors would thus be introduced into school curricula 

 previous to or along with the use of squared paper and 

 the idea of coordinates. 



In the discussion which followed, Sir Oliver Lodge, Dr. 

 Sumpner and others spoke as to the usefulness of vectorial 

 methods in physical work. Prof. Larmor said there could 

 be no doubt as to the extreme elegance of vectorial methods, 

 and attributed the slow progress they had made to the 

 want of uniformity in definitions' and notation, which 

 rendered it necessary for each writer who used vectors to 

 describe his notation and methods before his work could 

 be understood by his readers. Mr. Swinburne also referred 

 to this difficulty. Prof. Boltzmann pointed out that this 

 confusion would have been avoicled if Hamilton had accepted 

 Grassmann's methods and notation. The writer suggested 

 that the question of the possibility of introducing greater 

 uniformity into the notation and methods of vector algebra 

 was a suitable one to be considered by a committee of the 

 British Association. Prof. Henrici thought there would be 

 little difficulty in coming to some agreement between the 

 advocates of the various systems now in existence. His 

 communication was ordered to be printed in extenso in the 

 reports, so that those interested in the subject might be able 

 to consider the suggestions made m detail. 



Mr. Swinburne opened the discussion on the treatment 

 of irreversible processes in thermodynamics by pointing out 

 that so much attention was devoted in books on thermo- 

 dynamics to the consideration of the changes involved in 

 reversible processes, and so little to irreversible ones, that 

 there was a danger of the latter being overlooked, although 

 they are the only ones which really occur in nature. His 

 object was to bring them more prominently forward, and 

 to suggest a method of introducing the subject which would 

 not involve alteration or extension of fundamental ideas 

 on passing from reversible to irreversible changes. The 

 sketch of the method he proposed was necessarily brief, and 

 it was not easy at the time to see to what the proposals 

 made would eventually lead. This probably accounts for 

 the unsatisfactory nature of the discussion, which consisted 

 to a great extent of statements by the speakers that they 

 had been unable to understand what was proposed, or of 

 condemnation of any attempt to alter the definition of 

 entropy. Fortunately, copies of Mr. Swinburne's com- 

 munication were available, and a quiet perusal of his 



NO. 1773, VOL. 68] 



suggestions shows that they are by no means so drastic 

 as was supposed. 



He points out that, while the first law of thermo- 

 dynamics asserts that heat is a form of energy, the 

 second states that only a portion of a given supply of heat 

 is available for conversion into work, although energy 

 of other forms is wholly convertible. That part of a 

 supply of heat which cannot be converted into work during 

 a cyclic change of state of the body containing the heat 

 he proposes to call the " waste heat." It depends on the 

 temperature of the coldest available reservoir of heat of 

 large capacity, say that of the sea. Any process which goes 

 on in an isolated system involves in general an increase 

 of this " waste," and the quotient of this increase by the 

 temperature of the coldest available reservoir of heat Mr. 

 Swinburne defines as the increase of entropy of the system 

 during the process. 



A part of the system may decrease in entropy, but the 

 rest must increase by at least an equal amount. If the 

 increase is equal to the decrease the increase is said to 

 be "compensated," if it exceeds the decrease the excess is 

 the " uncompensated " increase of entropy. A reversible 

 change in an isolated system involves no increase of entropy 

 of the system, and any change in the entropy of any part 

 of the system must therefore be " compensated." When 

 irreversible changes occur there is an increase of entropy 

 of the system, and an uncompensated increase of entropy 

 of some part of it. So far as reversible changes are con- 

 cerned, it is evident that Mr. Swinburne's definition of 

 entropy leads to the same result as the one commonly used, 



i.e. I — = d<l>. For if in a Carnot cycle heat H^ i> 



taken in by the working substance at a temperature 0,, the 

 increase of entropy of the substance =H,/^i, and if at the 

 temperature of the coldest available reservoir 6„, H„ is 

 given up by the substance, H„ is Mr. Swinburne's waste 

 heat, and Hg/^o. according to his definition, the increase 

 of entropy of the substance when it took in H, from the 

 reservoir 0^. As temperatures are measured on the absolute 

 scale, the two quantities are identical. 



From this point onwards Mr. Swinburne's treatment of 

 the equilibrium of isolated systems is much like those in 

 use at present, except that he objects to the use of some 

 of the names, e.g. " thermodynamic potential," now 

 commonly used. 



Prof. Perry, in the discussion which followed, stated that 

 engineers, while using the definition of entropy which con- 

 nected it with reversible changes, were quite aware 

 that most of the processes with which they had to deal 

 were irreversible, and that their theory was an approxim- 

 ation only. 



Prof. Larmor thought Mr. Swinburne's method was a 

 praiseworthy attempt to introduce simplification and pre- 

 cision into a part of the subject which had received little 

 attention, and was still somewhat obscure, and Mr. Boys 

 added that the ideas brought forward were well worthy of 

 careful consideration. 



Before stating his views as to the nature of the eman- 

 ations from radio-active substances. Prof. Rutherford gave 

 a short resumi of the known facts about radio-activity. 

 Substances which possess the property throw off material 

 which carries with it a positive electric charge. This 

 charged material can penetrate to some extent through 

 solids, is deviated in electric and magnetic fields, and 

 appears to consist of particles of matter of about twice the 

 weight of a hydrogen atom, moving with a velocity about 

 one-tenth that' of light. This is known as the a radiation, 

 and accounts for about 99 per cent, of the energy sent out 

 by a radio-active substance. Another kind of radiation, 

 known as the j8 or kathode ray, is also emitted. It is 

 negatively charged, more penetrative and more easily 

 deviated than the a radiation, and appears to consist of 

 particles of about one-thousandth the mass of the hydrogen 

 atom. A third kind of radiation, known as the 7, is more 

 penetrative still, but up to the present has not been 

 sufficiently studied to enable its properties to be definitely 

 stated. The matter which remains after the a radiation has 

 been thrown off behaves in the case of thorium and radium 

 like a gas of large molecular weight, diffuses, condenses at 

 low temperatures, may deposit itself on bodies with which 

 it comes into contact, and may again divide into a posi- 



