466 proceedings of the american acadexmy. 



Electromagnetics and Mechanics. 

 The Cmitinuous and Discontinuous in Physics. 



45. It has been customary in physics to regard a fluid as composed 

 of discrete particles (as in the kinetic theory) or as a continuum (as in 

 hydrodynamics) according to the nature of the problem under investi- 

 gation; it has been assumed that e\en if a fluid were made up of 

 discrete particles, it could be treated as a continuum for the sake of 

 convenience in applying the laws of mathematical analysis. For 

 example we introduce the concept of density which may have no real 

 exact physical significance, but which by the method of averages 

 yields apparently correct results. Provided that the particles in a 

 discontinuous assemblage are sufficiently small, numerous, and regu- 

 larly distributed, it is assumed that any assemblage of discrete 

 particles can be replaced without loss of mathematical rigor by a 

 continuum. 



However, when we investigate problems of this character in the 

 light of our four dimensional geometry, we are led to the striking 

 conclusion that in some cases it is impossible, except by methods 

 which are unwarrantably arbitrary, to replace a discontinuous by a 

 continuous distribution and vice versa. Especially we shall see that 

 this is the case with radiant energy, a conclusion which is particularly 



Hence 



O-P = -l-c^^'(/?) +\aR)') + {RfiR) + mR))- 



If <3> tp is to vanish regardless of the curvature of the (a)-curvc, then 



„' (R) + ~^ (R) = 0, Bf'iR) + 3/(i?j = 0. 



The integration of these equations determines <p and / as 



A B^ 



" ~ W ^ ~ R'' 



where A and B are constants. The expression for <^xp is 



o>^p = - 4 (ixc + — p-^ i>^^) - ^ix^- 



R^\ ' R I R 



The calculation of <0'*OxP = ~ O*0"P gives 



It therefore appears impossible to satisfy <(^«p = Oand<^«<(^p = with any 

 other form of potential, dependent only on 1 and w, than the one chosen. 



