32, 33] MOMENT OF MOMENTUM. 95 



Inserting these values in d'Alembert's equation we obtain 

 58) <? 



If U depends only on the mutual distances of the particles of 

 the system it is unchanged in the displacement, dU=0. 

 We then have 



As was mentioned in 11 the quantity within the parenthesis 

 is an exact derivative , so that 



or differentiating outside of the sign of summation 



Integrating we obtain 



60) ^rm r \y r z r -} = H X) an arbitrary constant. 



The expression m(y~ z^\=ymv z 2mv y is the moment of 

 momentum [42), 13] about the X-axis of the mass m, or it is the 



f Qf 



product of twice the mass by the sectorial velocity -~^ ( 8). The 



theorem consequently states that the moment of momentum of the 

 whole system with respect to the X-axis is constant. 



Under similar conditions for the other two axes we obtain 



dx dz 





dx 



The vector H, whose components are H x , H y , H z , is the resultant 

 moment of momentum of the whole system, and if the above equa- 

 tions 60) hold it is constant both in magnitude and direction. This 

 is the case for the solar system and we accordingly have an unvary- 

 ing direction in space characteristic of the system. This direction 

 was called by Laplace that of the Invariable Axis and the plane 

 through the sun perpendicular to it the Invariable Plane. It may 

 be defined as that plane for which the sum of the masses of each 

 particle multiplied by the projection of its sectorial velocity on that 

 plane is a maximum. Such a plane furnishes a natural plane of 

 coordinates for the solar system. 



