DYNAMICAL PRINCIPLES TO PHYSICAL PHENOMENA. 
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and it cannot be a function of the temperature coordinates u because these coordinates 
are kinosthenic, i.e., they only enter the expression for the kinetic energy through 
their differential coefficients. 
In order to see that { xx } may be a function of the strain coordinates w, it is con¬ 
venient to notice that the kinetic energy of a system in ordinary dynamics may be 
written in the form 
.(14) 
where M is the mass, x a coordinate fixing the position of the centre of gravity, k a 
radius of gyration, and 6 the angle made by a line fixed in the body with a line fixed 
in space. Terms of the type TVL 3 evidently cannot involve the w coordinates, but terms 
of the type M& 3 # 3 may; for take the simple case of a bar which is compressed at its 
middle and extended at its ends, rotating about an axis through its centre, it is easy to 
see that the moment of inertia of this rod about the axis of rotation is less than it would 
be if the rod were unstrained, and thus M/r may be a function of the strain components. 
These components will in general only enter through the expression for the alteration 
in the density, i.e., using Thomson and Tait’s notation through the expression 
da./dx-\-dfi/dy-\-dy/dz, and this expression will only occur raised to the first power ; 
if we employ the energy method of forming the equations of elasticity, we easily find 
that the presence of this term leads to the introduction of the so-called “ centrifugal 
force ” into the equations of elasticity for a rotating elastic solid. 
Collecting our results we see that {xx} may be a function of the coordinates 
x, y and w, but not of u and probably not of z. 
§ 5. The next terms which we have to consider are the terms of the type {yy}y 2 , 
which is supposed to include terms of the form {y^y^Viy^ 
Now since the kinetic energy of a circuit carrying a current y is -|L y' 2 if L be 
the coefficient of self-induction of the circuit, we see that {yy} is a coefficient of 
self-induction; and since the coefficient of self-induction depends on the shape of the 
circuit {yy} will be a function of the geometrical coordinates which fix the shape of the 
circuit : since there is a mechanical force between two circuits conveying electric 
currents, {y^y*} must be a function of the geometrical coordinates which fix the 
relative position of the circuits, for if it were not so, we see by Lagrange’s equations 
that there would be no mechanical action between the circuits. Let us suppose that 
we have two circuits, and that the current in one circuit is y L and the current in the 
other y 2 . Let the kinetic energy of the system be 
.( 15 ) 
then if x be any coordinate fixing the position of one circuit with respect to the other, 
Lagrange’s equation for x shows that there will be a force tending to increase x 
equal to 
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