754 
Motion of a system of incoherent material points. 
§ 4. Let us now, following Ersrein, consider a very large number 
of material points wholly free from each other, which are moving 
in a gravitation field in such a way that at a definite moment the 
velocity components of these points are continuous functions of the 
coordinates. By taking the number very large we may pass to the 
limiting case of a continuously distributed matter without internal 
forces. 
Evidently the laws of motion for a system of this kind follow 
immediately from those for a single material point. If o is the 
density and de dy dz an element of volume, we may write instead 
of (8) 
— 0 V-S(abdgar va vy da dy de. se et EN 
If now we wish to extend equation (3) to the whole system we 
must multiply (9) by dé and integrate with respect to #, y, 2 and 7. 
In the last term of (8) we shall do so likewise after having 
replaced the components A, by Ka de dy dz, so that in what follows 
K will represent the external force per unit of volume. 
If farther we replace de dy dz dt by dS, an element of the four- 
dimensional extension a,,...@,, and put 
ON ES Daar nn VR Prine | Eer (10) 
L=— V (ab) CAs At a ay oe I PISS (11) 
we find the following form of the fundamental theorem. 
Let a variation of the motion of the system of material points be 
defined by the infinitely small quantities dr,, which are arbitrary 
continuous functions of the coordinates within an arbitrarily chosen 
finite space S, at the limits of which they vanish. Then we have, 
if the integrals are taken over the space S, and the quantities 
Jab ave left unchanged, 
J | Las + [ 20) Kara. dS=0 ot vn 
For the first term we may write 
je 
| dL. dS, 
if dL denotes the change of L at a fixed point of the space S. 
The quantity LdS and therefore also the integral /LdSis invariant 
when we pass to another system of coordinates. *) 
1) This follows from the invariancy of ds?, combined with the relations 
o' 0) 1 
sp, dS =—dS. 
da 4 de, P 
