318 
MR. J. J. THOMSON ON SOME APPLICATIONS OP 
,dL- , . PM- ■ . ,PN - 0 
dx 
■dx 
(16) 
Now we see that dh/dx and dddfdx must vanish, otherwise there would be a force 
between the two circuits when the current in one of them vanished, thus we see that 
the coefficients of self-induction are independent of the position of the other circuits 
in the neighbourhood. Thus the force tending to increase x reduces to 
PM • • 
(17) 
We see also from Lagrange’s equation for the coordinate y 1 that 
d PT_PT 
dt dy x dy x 
external force of type y 1 
But 
— 0, as we shall see directly, and therefore 
dy l 
(18) 
—(Ly 1 +My 3 )= external force of type y x .(19) 
Cv 6 
so that the term d(Ly x -\-My 2 )/dt will produce the same effect as an external electro¬ 
motive force equal to 
-f(L^+My s ).(20) 
and thus there is an electromotive force of this mao-nitucle acting round the circuit 
through which the current y x flows. This expression expresses the law of the induction 
of currents, either by the motion of neighbouring circuits which convey currents or by 
the alteration in the magnitude of the currents flowing through these circuits. This 
example is given in Maxwell’s ‘ Electricity and Magnetism,’ vol. ii., chap. 7; it 
illustrates the power of the method very well, as the existence of a mechanical force 
between two circuits showed that there was a term of the form My x y 2 hi the expression 
for the kinetic energy, and from this the law of induction followed at once by 
Lagrange’s equations. There is no experimental evidence to show that {yy} is ever 
a function of* y the electrical coordinates, and when the system consists of a lot of 
conducting circuits it certainly is not, for if it were the coefficients of self and mutual 
induction in a system of circuits would depend upon the length of time the current 
had been flowing through the circuits; in any case it would involve the existence of 
electromotive forces which would not be reversed if the directions of all the electric 
displacements in the field were reversed. 
There seems also good reason for believing that {yy} is not a function of the 
