SIXTH ORDINARY MEETING. 47; 



Supposing jo to be constantly uniform if the radius of the sphere 



T 



be originally a and become a - da, do will evidently be - da, and the 



total amount of work done on account of the contraction, will con. 



sequently be - M 2 — , where M — pa 3 , the mass of the sphere. 

 D a" o 



Integrating this expression between the limits a and b we get as, 



the amount of work done by a spherical mass M of radius a (supposed 



uniform) contracting to a uniform sphere of radius b, - M 2 ( — - 1 . 



5 \b aj 



Applying these formulae to the case of the sun whose radius is 

 433,200 miles and whose mass is 4 (10j 30 lbs., the amount of work 

 done, or in other words, tne quantity of heat generated, by a con- 

 traction of 1 foot in the radius of the sun (supposed uniform) will be 

 found to be represented by — 



3 16 (10 60 



5 (433200) 2 (5280) 2 



The unit of force used here obviously is the attraction of unit mass 

 on unit mass at unit distance ; so that the attraction of the earth on. 

 unit mass at its surface would be represented by — 



4 1 



(10) 2 ' 



33 v ' (400) 2 (5280) 2 



4 



multiplied by the mass of the earth = — ,10) 60 of these units. 



oo 



Now this force will cause 1 lb. to move through - = 16'1 ft. per 



second. 



Therefore a contraction of 1 foot in the sun's radius will generate 

 a quantity of heat equivalent to — 



3 16 (10) 60 x 33 x (4000) 2 x (5 280) 2 x 16T 

 5 " ~~ (433200) a x 4 x (10) 26 



= (10) 33 foot-pounds. 



If account were taken of the fact that the sun must become denser 

 as its centre is approached, this quantity would be considerably 

 larger. 



Accordingly a yearly contraction of 10 feet in the sun's radium 

 would be amply sufficient to sustain its heat at the present rate of. 

 radiation. 



