WILSON AM) LKWIS.— KKI,.\TI\ ITV. 493 



due to the Maxwell strains and the rate of ehange of the Poynting 

 vector; the second is Poynting's theorem. ®° 



Mechanics of a Material System, and Gravitation. 



58. The mechanics of a partick^ wliich we have treated in restricted 

 cases in § 21 and § 36 can now be completely generalized. If Wq is 

 the mass of a particle, and w the unit tangent to its locus, then 



7?ZoW = /// (V + k4) 



is the vector of extended momentum, whose projections on any chosen 

 space and time are niv, the momentum, and m, the mass or energy. 

 If we consider any number of such vectors, we may state the laws of 

 conservation of momentum, mass and energy in a single theorem as 

 follows. The sum of all the vectors of extended momentum is constant, 

 that is, the sum of all such vectors cutting any unclosed and continu- 

 ous three dimensional (7)-spread is independent of the (7)-spread 

 chosen. This law is, however, true only when we state that wherever 

 there is energy there is a vector of extended momentum, whether, 

 or not this energy is associated with that which is ordinarily known as 

 a material system. Thus in § 51 we have discussed the vector dg 

 which we have identified with the vector of extended momentum of 

 radiant electromagnetic energy. A Hohlraum obeys all the laws of 

 a material system, and must be treated as such. We shall mention 

 presently another form of radiant energy to which also we must assign 

 an extended momentum. 



Just as the discrete locus of an electric charge was replaced by a 

 continuously distributed field of density vectors, we might regard a 

 material system as a continuum. Thus if we have a small (5)-tube 

 parallel to and comprising one or more (6)-lines of which the resultant 

 vector is W()W,we may replace this vector by the expression ((/S>^MoW)*w, 

 where f/S is the intersection of the tube with any planoid, and /xqW 

 is the vector of the distributed field. If (/S is taken perpendicular 

 to w, this reduces to jUoWf/S, and therefore ^uq is the density as it appears 

 to an observer at rest with respect to the system. It must, however, 



60 In case there is electricity present, these equations become respectively 



V*^s - i.exh = -1 ,rp(e + vxhK V*exh + ^, T, = - -irpv-e. 



ol dt 



Note that if v is small, the second equation is corrected by the .small term 



— 4n^pV»e. whereas the first has the large correction 47rp(e + Vxh), approxi- 

 mately 4;rpe. 



