Hence we have the following fundamental theorem due to Helmholtz, which is the basis of the theory of sun heat. _If the sun he regarded as a homogeneous sphere of mass_ M _pounds and radius a feet, then the foot-pounds of energy rendered available for sun heat by the contraction of the solar material from, an infinite distance is_ 16 a M. § 4. EVALUATION OF THE SUN HEAT GIVEN OUT IN CONTRACTION. The number of foot-pounds of work given out in the contraction from infinity is 16 _a_ M. As 772 foot-pounds are equal to one unit of heat, _i.e._ to the quantity of heat necessary to raise 1 lb. of water 1° Fahrenheit, we see that 772 M is the work required to raise a mass of water equal to the mass of the sun through 1° Fahrenheit. Hence the number of globes of water, each equal to the sun in mass, which would be raised 1° Fahrenheit by the total heat arising from the contraction, is (16 _a_)/772, but _a_, the radius of the sun in feet, is 2,280,000,000, and hence we have the following theorem:— _The energy liberated in the contraction of the sun from infinity to its present dimensions would, if turned into heat, suffice to raise 47,000,000 globes of water, each having the same mass as the sun, through 1° Fahr._ It is found by experiment that 1 lb. of good coal may develop 14,000 units of heat, and is therefore equivalent to 14,000 × 772 foot-pounds of work. A mass of coal equal to the sun would therefore (granted oxygen enough) be equivalent to 14,000 × 772 × M foot-pounds of work. But we have (16 _a_ M) 16 × 2,280,000,000 –––––––––––––– = ––––––––––––––– = 3,400. 14,000 × 772 × M 14,000 × 772 Hence we see that _The energy liberated in the contraction of the sun from infinity to its present dimensions, is as great as could be produced by the combustion of 3,400 globes of coal, each as heavy as the sun._ We may speak of 3,400 in this case as the coal equivalent. § 5. ON THE FURTHER CONTRACTION OF THE SUN AND THE HEAT THAT MAY THUS BE GIVEN OUT. Let us suppose the sun contracts to the radius _r_, and then, as already proved, § 3, the energy it gives out is ⅗ (ε M^2)/_r_, but we have ε M/_a_^2 = 27, whence on contraction to the radius _r_ the total energy given out from the commencement is 16 M (_a_^2/_r_) The average density of the sun at present is 1.4. Let us suppose it condenses until it has a density ρ. _r_^3 ÷ _a_^3 = 1.4 ÷ ρ, whence the energy becomes 14 _a_ M · ∛ρ; but the coal equivalent of 16 _a_ M has been found in § 4 to be 3,400, and hence the coal equivalent in this case is 3,000 ∛ρ.