580 PROCEEDINGS OF THE AMERICAN ACADEMY. 



To form an idea of the extent to which the inertia of a large mass 

 is diminished by the gravitational energy, we may calculate the ratio 

 vig/m for a sphere of uniform density p and radius a. In such a case, 

 at all external points whose distance is r the vector g has the intensity 

 km/r^, while at internal points its intensity is kmr/a^. The energy 

 is now 



1 C^ 



— nigC^ = — -— I i-n-r^g^dr, 

 Stt/c Jo 



which is readily shown to be 



Sm^'k 



Hence we see that 



For a sphere of homogeneous density and of size and mass equal to 

 that of the sun, we have 



m = 2.0X 10'^ a = 7.0 X 10'°, k = 6.5 X 10"^ c^ = 9.0 X 10'", 



17) 



SO that -^ = 1.2 X 10-' • 



m 



The inertia of the sun is therefore diminished by about one part 

 in a million by the gravitational energy it possesses. 



Consequences of Negative Inertia. — Small as it is, this minute 

 diminution of inertia shows clearly the way to the ultimate condi- 

 tion of the universe. For we may imagine the sun and all the rest 

 of the stars radiating their heat away as they drift through space, 

 and having the supply renewed only by occasional collisions, each one 

 of which combines the masses of the colliding bodies into a mass as 

 great as both together, but with an inertia, after the heat of collision 

 is lost,!^ not as great as the sum of those of the original masses. 

 Thus a time will come when one of the masses formed by this process 

 will have actually less inertia than one of the parts of which it is made 

 up, and will readily be accelerated to enormous velocities by the least 

 attractions. And at last there will be formed a tremendous mass, of 

 the order of 10® times the mass of the sun, whose inertia will be nega- 



" Because heat is a form of motion of bodies with electromagnetic mass, 

 we may consider it as electromagnetic energy, with inertia like that of any 

 other such energy. 



