66 Mr. J. J* Thomson. On some Applications of [Jan. 8, 



heading the nature of the boundary is determined, and is shown to be 

 either plane or spherical. And by the application of Green's 

 theorem it also becomes clear that inasmuch as the statical conditions 

 of the crystallising agent are now understood, the force functions 

 derived in the preceding chapter can be independently deduced 

 without aid of the assumption from any one of the primitive forms 

 of the systems under consideration. 



IV. " On some Applications of Dynamical Principles to Physical 

 Phenomena." By J. J. Thomson, M.A., F.R.S., Fellow 

 of Trinity College, and Cavendish Professor of Physics in 

 the University of Cambridge. Received December 16, 

 1884. 



(Abstract.) 



In this paper an attempt is made to apply dynamical principles 

 to study some of the phenomena in electricity, magnetism, heat, and 

 elasticity. The matter (including, if necessary, the ether) which takes 

 part in any phenomenon is looked upon as forming a material system, 

 and the motion of this system is investigated by means of general 

 dynamical methods, Lagrange's equations being the method most 

 frequently used. To apply this method, it is necessary to have 

 coordinates which can fix the configuration of the system, so in 

 the first part of the paper coordinates are introduced which fix the 

 configuration of the system, so far as the phenomena we are consider- 

 ing are concerned, i.e., we introduce coordinates which can fix the 

 geometrical, the electrical, the magnetic, the heat, and the strain 

 configuration of the system ; we call these, coordinates of the x, y, z, 

 u, and w types respectively. Some of these coordinates only enter 

 the expression for the kinetic energy through their differential 

 coefficients, and may be called gyroscopic coordinates, as such 

 coordinates are of frequent occurrence in problems about gyro- 

 scopes. 



The terms which involve the x, y, z, u, and w coordinates in the 

 expression for the kinetic energy will be of fifteen different types. 



There will be those terms which are quadratic functions of the 

 differential coefficients of the x coordinates, and corresponding terms 

 for the y, z, u, and w coordinates ; so that there are in this set terms 

 of five different types, all of which may exist in actual dynamical 

 systems. There are ten types of terms involving the products of 

 differential coefficients of two coordinates of different kinds. These 

 types are considered in order, and it is shown that we have 

 experimental evidence for the existence of only two of them, viz., 



