312 
MR,. J. J. THOMSON ON SOME APPLICATIONS OP 
potential energy of the original system, and two configurations are not supposed to 
coincide unless the configuration of these systems coincide also. 
We shall, however, in the following investigations sometimes make use of the 
potential energy in the usual way, as the analysis is the same, and the use of the 
ordinary method saves a good deal of explanation. 
§ 2. Let us now go on to consider the various terms in the expression for the kinetic 
energy of a material system, taking into account the geometric, electric, magnetic, and 
thermal conditions of the system. We are dealing with five sets of coordinates—the 
sets we have denoted by the letters x, y, z, u, w. 
The kinetic energy T will be of the form 
T=-|{[x 1 cr 1 ]a: 1 3 +2[x 1 cr 3 ]a; 1 a: 2 + . . . 
+ [TO]2/i 3 +2[2/ 1 2/ 3 ]y 1 ^ 
+ 2 [xy]xy+ . . . 
+ ... } 
where the quantities denoted by [aqaq], jjqyJ, [x±y{\, may he functions of the co¬ 
ordinates x, y, z, u, iv. 
The various terms in T can be divided into fifteen types; there are five sets, one 
corresponding to each of the coordinates x, y, z, u, w, where the terms are of the same 
character as 
f Y 1 'Y "1 y ^ _ 1 _ Q | ry* ry> | /y> ry> 
I j| lA'p 2 
where each term involves the squares of the velocities of the coordinates of one kind, 
or the product of two velocities of the same kind: it is evident that each of these five 
types can exist in actual dynamical systems. 
There are ten sets of the type 
[xy\xy 
involving the product of the differential coefficients of two coordinates of different 
kinds : we can see, however, that some of these can not exist in any actual material 
system. Thus, for example, since the u coordinates only enter through the tempera¬ 
ture, there can be no terms involving the product of the differential coefficient of u 
and the differential coefficient of any of the other coordinates : this consideration 
reduces the ten sets to six. To determine whether terms of the type of any particular 
set exist or not we must determine what the consequences would be if terms of this 
type did exist; if these are contrary to experience we conclude that terms of this type 
do not exist. We can determine these consequences in the following way. Suppose 
we have a term in the kinetic energy equal to 
(V)V 
