74 PROFESSOR W. THOMSON ON THE 



of the Sun being about 1^ times that of water, the matter in a pyramidal por- 

 tion from his centre to a square foot of his surface is about 



I x 441,000 x 5280 x 1£ x 64 = 62,100,000,000 lb. 

 and the whole annual addition of meteoric matter to the Sun would there- 

 fore be 



1900 1 



62,100,000,000 32,400,000 



of his own mass. In about six thousand years the Sun would therefore be aug- 

 mented by 5(^0 i Q mass from extra-planetary space. Since the time occupied by 

 each meteor in falling to the Sun from any distance would be much less than 

 the periodic time of a planet revolving at that distance, and since the periodic- 

 times of the most distant of the planets is but a small fraction of 6000 

 years, it follows that the chief effect on the motions of the planetary system pro- 

 duced during such a period by the attraction of the matter falling in would be 

 that depending simply on the augmentation of the central force. To determine 

 this, let M be the Sun's mass at any time t, measured from an epoch G000 years 

 ago ; w the Earth's mean angular velocity, and a its mean distance at the same 

 time ; and 2 h the constant area described by its radius vector per second. Then 

 we have — 



M 



from which we deduce, 



and 



« x* — , (centrifugal force) 



u a 2 — h ; (equable description of areas) 



h 2 

 M 



M 2 



Now, if M denote the mass of the sun at the epoch from which time is reckoned ; 

 since the annual augmentation is about mtmoOo of tne mass i tse l£ we have 



M = M « ( X + 323o'wo) ' 



and M2 = M « 2 ( 1+ 32JP00)- 



Hence, if n and a, denote the angular velocities at the epoch and at the present 

 time, T ; the angular velocity, which is uniformly accelerated during the interval, 

 will have a mean value, n, expressed as follows : — 



a = § (a + a,) = a T fl-J.5d3W/i- ^L__\ • 



V / I n r J V 32,400,000/ ' 



and if © denote the angle described in the time T, we have 



© = a T (t - 11 V 



T V 32,400,000/ 



