320 
MR. J. J. THOMSON ON SOME APPLICATIONS OF 
elasticity for copper. Thus, supposing that we can measure a coefficient of elasticity 
correct to one part in a thousand, in order that we may be able to detect the 
alteration in the coefficient of elasticity produced by a current of 10 amperes—whose 
value in absolute measure is unity—the quantities A and B must be at least TO 11 /!000 
or 10 8 . Now we can produce strains in a copper wire greater than 1/10000 without 
breaking the wire, and if we produce strains of this magnitude the alteration in the 
coefficient of self-induction will be between 1 and 10 per centimetre of wire in the 
coil; supposing A to be 10 8 ; this is a very large change in the coefficient of self-induc¬ 
tion, and ought to be easily detected, much more easily than the corresponding changes 
in the elasticity of a copper wire produced by the passage of a current of electricity. 
To sum up, we see that {yy} is a function of the coordinates of the type x, but not a 
function of the y, z, or u coordinates, while it is doubtful whether it is a function of 
the strain coordinates w or not. 
§ 6. Let us now consider the terms of the type { zz}z 3 , these are the terms which 
express the magnetic energy of the system. In § 1 we discussed a method of fixing 
the magnetic configuration by means of coordinates. To fix the intensity of magneti¬ 
sation we use two coordinates, y and £, of which y is a coordinate of a geometrical 
kind, and £ a kinosthenic coordinate, and they are chosen so that the intensity of 
magnetisation is y£, where £ is the momentum of the type £. Since £ is a kinosthenic 
coordinate f is constant, it is therefore convenient to use ^ instead of £ in the expres¬ 
sion for the kinetic energy. Now Bouth (‘ Stability of Motion,’ p. 62) has shown 
that if 6 be a coordinate and clTJdd=a, then if we eliminate 6' from the expression 
T'=T— a& .(22) 
we may use Lagrange’s equations for all the other coordinates if we use the function 
T' instead of T. Let us apply this theorem to the case we are considering: suppose 
we have a term of the form A£ 3 in the expression for the kinetic energy T, we must 
eliminate £ by the use of the equation 
f=f. 
as 
. . . (23) 
and use T' instead of T where 
T=T-£i . 
. M 
thus the term containing in T' is 
ill 
2 a . 
.(25) 
this is of the same magnitude as the corresponding term in T but of opposite sign. 
[This change of sign is instructive if, as we have done in this paper, u r e regard 
potential energy as kinetic energy due to kinosthenic coordinates, for it explains at 
once why the Lagrangian function is T — V and not T-f-V ; for as we saw above, the 
