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MR. J. J. THOMSON ON SOME APPLICATIONS OF 
products of the differential coefficients of coordinates of different kinds, and see 
whether they exist or not. 
1. Terms of the type [xy}xy : we have just seen that these terms do not exist. 
See also Maxwell’s ‘ Electricity and Magnetism/ vol. ii., chap. 7. 
2. Terms of the type {xz}xz : z is a coordinate fixing the magnetic configuration; 
we can see just as in the last case that the existence of this term would require the 
production of a magnetic force by a body whose velocity is changing depending on the 
rate of change of the velocity, as well as other similar effects. As these have not been 
observed, we conclude that this term does not exist. 
3. Terms of the type {xu}xu : u is a coordinate helping to fix the temperature ; 
as the “As” only enter the expression for the kinetic energy through the temperature, 
i.e., through terms of the type [uu}u z , we see that these terms do not exist. 
4. Terms of the type {xiv}xw : iv is a' coordinate helping to fix the strain con¬ 
figuration of the system ; these may exist in a vibrating solid which has also got a 
motion of translation, for the velocity of any point in a small element of the solid 
equals the velocity of the centre of gravity of that element plus the velocity of the 
point relatively to the centre of gravity. This latter velocity will involve w, so that 
the square of the velocity, and therefore the kinetic energy, may involve the 
product xiv. 
5. Terms of the type {yz}yz: the production of a magnetic field by an electric 
current such that the direction of magnetic force is reversed when the direction of 
the current is reversed shows that terms of this type must exist. 
6. Terms of the type {yu}yu : the reasoning given when we were discussing 
(3) terms of the type { xu}xu shows that terms of this type do not exist. 
7. Terms of the t} T pe {yiv}yw : by reasoning in the same way as we did about the 
general term [X/x]\y, we can see that if this term exists a varying current will 
produce strains differing from those produced by a steady current, and also an 
electromotive force could be produced by merely altering the state of strain of a 
body. As these effects are not known to occur, we conclude that terms of this type 
do not exist. 
8. Terms of the type {zu}zu : the reasoning given when we considered the 
term (3) shows that terms of this type do not exist. 
9. Terms of the type {ziv}zw : terms of this type would correspond to effects 
analogous in character to those due to the terms of the type {yw}yw, and as these 
have not been observed, we conclude that terms of this type do not exist. 
10. Terms of the type \uiv\uw : the reasoning given when we considered the 
term (3) shows that terms of this type do not exist. 
We thus see that all the terms which occur in the expression for the kinetic energy 
of any real material system are of one or other of the following types :— 
