752 
occurring in fhe real motion in the way defined by the infinitely 
small quantities Òr,dy,dz. If, in the varied motion, the position 
v + dv,y + dy,z + dz is reached at the same time ¢ as the position 
vy.c in the real motion, we shall have the equation 
sf La +f (Kida + Ki dy -| Kidz) d= 0, Me tad 
L being the Lagrangian function and the integrals being taken over 
an arbitrary interval of time, at the beginning and the end of 
which the variations of the coordinates are zero. A is supposed to 
be a force acting on the material point beside the forces that are 
included in the Lagrangian function. 
4 
§ 2. We may also suppose the time ¢ to be varied, so that in the 
varied motion the position « + dv, y+ dy, z dz is reachedgat the 
time ¢-+ dt. In the first term of (1) this does not make any difference 
if we suppose that for the extreme positions also d¢ — 0. As to the 
second term we remark that the coordinates in the varied motion at 
the time ¢ may now be taken to be « + dw — v,dt, y + dy -— v‚dt, 
z+ dz — v,0t, if V,,0,.0, are the velocities in the real motion. In 
the second term we must therefore replace dv,dy,dz by da—v,dt, . 
* dy—v,0t, dz —v,dt. In the equation thus found we shall write 
Vy ly@,,e, for a,y,c,t. For the sake of uniformity we shall add to the 
three velocity components a fourth, which, however. necessarily must 
have the value 1 as we take for it = We shall also add to the 
an, 
three components of the force A a fourth component, which we 
define as 
Ky = he, Kot my tothe, Badsck noes WS 
and which therefore represents the work of the force per unit of 
time with the negative sign. 
Then we have instead of (1) 
a { Lat fe (a) Ka hey dt =D. el et ceding Saen 
and for (2) we may write *) ‘ 
1) In these formulae we have put between parentheses behind the sign of 
summation the index with respect to which the summation must be effected, which 
means that the values 1, 2,3, 4 have to be given to it successively. In the same way 
two or more indices behind tbe sign of summation will indicate that in the 
expression under this sign these values have to be given to each of the indices. 
s(ab) f. i. means that each of the four values of a has to be combined with 
each of the four values of b. 
