494 PROCEEDINGS OF THE AMERICAN ACADEMY. 



be borne in mind that when the system in question embraces any 

 energy which is moving with the velocity of Hght, this method fails 

 completely. x\nd this is an essential difference between a system of 

 electric charges and a system of matter or energy. Indeed a consid- 

 eration of the properties of a Hohlraum shows that it may be unsafe 

 in any case to assume that a material system is not composed wholly 

 or in part of energy moving with the velocity of light. 



59. In the study of hydrodynamics cases are considered in which 

 the different portions of the fluid exert forces upon one another, and 

 these forces may be themselves due to a flow of energy with the 

 velocity of light. In fact it is only when we consider a fluid devoid of 

 such mutual forces that we are able to obtain from our continuously 

 distributed field and the law of extended momentum the known equa- 

 tion of hydrodynamics. Let us consider a continuously distributed 

 field divided into infinitesimal tubes in each of which the extended 

 momentum is now written as (rfS^Mow)*w. Then our conservation 

 law leads to the equation 



/ 



(f/SxAtow)*w = const. (150) 



Or if we consider a portion of the field composed of a number of 

 adjoining tubes and cut off by two different planoids, then since none 

 of the vectors of extended momentum cut the boundary tube the 

 integral of our vector over the whole three dimensional boundary of 

 this four dimensional region is merely the integral over the two planoids 

 namely, 



— / (f/S*«MoW)w = = — / rfS*«('/ioWw), 



by definition of the dyadic moWW. Now by the application of (65) 

 we may con^'ert this triple integral into a quadruple integral. Thus 



Jrf*S-(MoWw) = J*^S*0-(moWW) = 0. 



Hence 



<>-Uww) = 0. (151) 



If now we set w = (v + k4)/Vl — v~ and m = Mo/(1 — ^J^) by (88), 

 this gives by expansion®^ 



0-[m (V + k4) (V + k4)] = iO'fX (V + k4)] (V + k4) 



+ L u (v -f k4) .0](v + k4) = 0, 



61 If ab is a dyadic, evidently <0* (ab) = (O'a)b + (a«0)l>- 



