NA TURE 



585 



THURSDAY, OCTOBER 18, iJ 



APPLICATIONS OF DYNAMICS TO PHYSICS 

 AND CHEMISTRY. 



Applications of Dynamics to Physics and Chemistry. 

 By J. J. Thomson, M.A., F.R.S., Cavendish Professor 

 of Experimental Physics, Cambridge. (London : 

 Macmillan and Co., 1888.) 



THIS is one of the most original books on mathe- 

 matical physics which has appeared for a long 

 time. Prof. J. J. Thomson has elaborated a method of 

 very wide scope, and has applied it to a large number of 

 problems of very different kinds. A reader of the work 

 must perforce be struck not only with the mathematical 

 ability of the author, but with the wide extent of learning 

 which enables him to illustrate his theme by recent 

 researches in nearly every branch of physics and physical 

 chemistry. 



The method employed is so essentially mathematical 

 that it is not easy to describe it without the use of 

 symbols. As, however, it is a matter of considerable 

 importance that those who are studying by means of 

 experiment the phenomena discussed by Prof. Thomson 

 should have some idea as to the progress already made in 

 their theoretical explanation, it may be well to give an 

 account of the general principles which he has used. 



In ordinary dynamics it is necessary to specify the 

 positions of the members of a system of bodies of which 

 the movements or mutual actions are under considera- 

 tion. This is done by means of co-ordinates, which 

 define their positions at a given time with respect to 

 certain given lines or surfaces. If the system is in motion, 

 the values of these quantities change with the time, and 

 thus the co-ordinates may be regarded as possessing 

 velocities. 



The difference between the kinetic and potential 

 energies of the system (which is called the Lagrangian 

 function) can be expressed in terms of the co-ordinates 

 and their velocities, and if this is done the magnitude of 

 the force which is acting on the system and tending to 

 increase the value of any particular co-ordinate can be 

 deduced from it. If no such force is acting, it follows 

 that a certain relation between the co-ordinates and their 

 velocities must be satisfied. 



This is a perfectly general dynamical method, which 

 could be directly applied to the complex system of atoms 

 and ether by which the physical phenomena displayed 

 by any given body are produced, if it were not for 

 difficulties which Prof. Thomson has attempted, as far as 

 may be, to overcome. 



In the first place, the dynamical method presupposes 

 a knowledge of the relative positions of the members of 

 the system, i.e. of its geometry, and we cannot at present 

 express " such things as the distributions of electricity 

 and magnetism, for example," in terms of the relative 

 positions or movements of atoms and ether. 



In the next place, even when we can express certain 



physical states in terms of quantities which completely 



describe all that we can observe, it is certain that in 



general they would not suffice to describe completely the 



Vol. xxxviii. — No. 990. 



state of the body if we had the power of noting every 

 detail of its molecular structure. 



Using theorems due to Thomson and Tait and to 

 Larmor respectively, Prof. J. J. Thomson shows that the 

 second difficulty may be overcome if the co-ordinates of 

 the values of which we are ignorant do not enter into the 

 expressions for the kinetic energy of the system. It then, 

 however, becomes necessary to modify the Lagrangian 

 function, but this new form is such that when it is ex- 

 pressed in terms of any variable quantities and their 

 "velocities" they satisfy the mathematical condition to 

 which a true geometrical co-ordinate and its velocity are 

 subject. If L' is the modified Lagrangian function, and q 

 any one of a series of quantities, q 1} q 2 , . . . , in terms of 

 which and of their velocities it can be expressed, then — 

 d dU_ dV^ 

 dt dq dq 



The term co-ordinate is thus used by Prof. Thomson in 

 the generalized sense of any quantity in terms of which and 

 of its velocity the modified Lagrangian function can be 

 expressed, and he assumes that as far as the phenomena 

 under consideration are concerned the state of a body 

 may be described by four different types of co-ordinates. 

 These specify (1) the position in space of any bodies of 

 finite size which may be in the system ; (2) the strains in 

 the system ; (3) its electrical, and (4) its magnetic state. 



The most general expression for the Lagrangian 

 function is then formed. It may contain terms of various 

 kinds. Prof. Thomson goes through them one by one, 

 determines what the physical consequences of the exist- 

 ence of each would be, and if these are found to be 

 contrary to experience concludes that the term in question 

 does not exist. 



Thus, for instance, it can be shown that if there were 

 a term containing a product of the velocities of a geo- 

 metrical and an electrical co-ordinate, an electrical 

 " current would produce a mechanical force proportional 

 to its square, so that the force would not be reversed if 

 the direction of the current was reversed." As this and 

 other similar deductions are all opposed to experience, 

 no such term can exist. 



A similar method is applied to the coefficients of the 

 terms which are shown to be possible. Thus a term 

 exists which contains the squares of the velocities of the 

 geometrical co-ordinates. It corresponds to the expres- 

 sion for the ordinary kinetic energy. Prof. Thomson 

 inquires whether the kinetic energy depends only on the 

 geometrical co-ordinates, or whether it also varies with 

 the electrical state of the various members of the system. 

 The answer is given by means of an investigation of his 

 own {Phil. Mag., April 1881), in which he has shown 

 that the kinetic energy of a small sphere, of mass ;;;, and 

 radius a, charged with e units of electricity, and moving 

 with a velocity v, is — 



2 fjLl'- 

 15 



where fi is the magnetic permeability of the dielectric 

 surrounding it. The effect of the electrification is there- 

 fore the same as if the mass had received an increment, 

 which, however, numerical calculation shows is too small 

 to be observed. It is, nevertheless, important that the 

 mutual relationship between ordinary kinetic energy and 



c c 



f: 



m + 



a ) 



