46 PROCEEDINGS OF THE CANADIAN INSTITUTE. 



sun was 4 tt(433) 2 (10) 6 (5280) 2 5,500,000 foot-pounds per second, or 

 (10) 34 foot-pounds per annum. It therefore follows that if the theory 

 of the origin of solar heat under examination were the true one, the 

 energy of the sun would be completely exhausted in 3,680 years, 

 while we know that the quantity of heat radiated from the sun has 

 been practically as great as at present for millions of years. The 

 theory of combustion or chemical combination, therefore, falls to the 

 ground, and it is now generally supposed that the perennial fountain 

 whence flow the vast energies of the solar system, is the potential 

 energy of gravitation which is converted into kinetic energy by its. 

 mass moving towards the centre of inertia of the solar system, and 

 thence into heat by a mechanism indicated by the physical constitu- 

 tion of the fiery ruler of the day. 



The following investigation will show that this now generally 

 accepted hypothesis predicates a cause known to be a vera causa 

 amply capable of producing the results it is supposed to explain, and 

 that therefore it is not inconsistent with the axiom that the cause 

 must be equal to the effect. 



Let p represent the density at distance r from the centre of a 

 spherical mass, supposed equally dense at equal distances from the 

 centre. The elemental mass, therefore, between the spherical surfaces 

 whose radii are r and r + d r, is p 4 iz r dr. 



Taking proper units of force, <fcc, and remembering the theorem 

 that the attraction of a spherical shell on an internal particle vanishes, 

 it follows that the force acting on this elemental mass is measured by 

 the quantity — 



4 it p r dr fl 4 iz p r 2 dr. 



assuming of course the Newtonian law of gravitation. The work 

 done by this elemental mass moving through an infinitesimal dc, will 

 consequently be — 



4 7T p r dr / £ 4 7r p r dr do. 



Integrating with respect to dr we get as the total work done — 

 / \ 4 Tip dr.f r 4:7: pr 2 dr. > dr. 



a formula which will be found to be of considerable use in solving 

 certain important classes of problems. 



