DYNAMICAL PRINCIPLES TO PHYSICAL PHENOMENA. 
313 
where X and fx may be any of the five kinds of coordinates which we are considering. 
We have by Lagrange’s equations 
d dT < 2 T 
dt dX dX 
external force of the type X 
( 6 ) 
If instead of T we consider only the term (Xp,)Xju, we see that it requires the existence 
of a force of the type X equal to 
(\ [ x)ix+~{\ii)ix z +t^(\fx)ixv . .( 7 ) 
and a force of the type [x equal to 
— M^+ (X/r) w 
( 8 ) 
when v is a coordinate of any other type. 
Thus as it is clearer to take a definite case, suppose that X is the geometrical co¬ 
ordinate x, and jx then electrical coordinate y, then if the term (xy) xy occurred in the 
expression for the kinetic energy, the mechanical force produced by a varying current 
would be different from that produced by a steady one; this is shown by the existence 
of the term (xy)y in equation (7), which now represents the force of type x, i.e., the 
d 
mechanical force. If [xy] were a function of y, the term y\xy\y^, which occurs in 
ay 
the expression for the mechanical force, shows that a current would produce a 
mechanical force proportional to the square of the intensity of the current, and which 
therefore would not be reversed if the direction of the current were reversed. Again, 
if we consider the expression for the force of type y, that is, the electromotive force, 
we see that the existence of this term implies the production of an electromotive 
force by a body whose velocity is changing depending on the acceleration of the body ; 
this is indicated by the term [xy]x : if [xy] were a function of x, equation (8) shows 
that a moving body would produce an electromotive force proportional to the square 
of its velocity. As none of these effects have been observed we conclude that this 
term does not exist. 
Another point worthy of attention is that the existence of a force of any type £ 
say, in any region, implies the existence in the expression for the kinetic energy of that 
region of terms involving £ or f, so that if we alter the values of £ or by external 
means there will be either an absorption or an emission of energy in the region which 
is the seat of the force of type £ This is one method of determining the seat of a 
force. The kind of energy absorbed or emitted will depend on the coordinates with 
which £ and £ are associated in the expression for the kinetic energy. 
§ 3 . We shall now go through the various types of the terms which involve the 
