SIXTH ORDINARY MEETING. 47 
Supposing p to be constantly uniform: if the radius of the sphere 
be originally a and become a - da, de will evidently be - da, and the 
total amount of work done on account of the contraction, will con_ 
3_.,da Ar 
sequently be 5 M? —, where M = 3? a3, the mass of the sphere. 
aw ‘ 
Integrating this expression between the limits a and 6 we get as 
the amount of work done by aspherical mass M of radius a (supposed 
: ; : : 3 He 
uniform) contracting to a uniform sphere of radius 6, 5 MV? é - :): 
a 
Applying these formule to the case of the sun whose radius is 
433,200 miles and whose mass is 4 (10)* lbs., the amount of work 
done, or in other words, tiie 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 ef lO: 
5 (433200)? (5280)? 
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 (10) 1 
33"! (400)? (5280)" 
ite 4 : 
multiplied by the mass of the earth = 33 (10)° of these units. 
/ 
Now this force will cause 1 lb. to move through 2 = G71 tt per 
second. 
Therefore a contraction of 1 foot in the sun’s radius will generate 
a quantity of heat equivalent to— 
3 16 (10)® x 33 x (4000)? x (5280)? x 16-1 
5 (433200)? x 4 x (10) 
= (10)* 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 radius 
would be amply sufficient to sustain its heat at the present rate of 
radiation. 
