CASE XIV. 121 



where the first term comes from the tides raised by the moon and the second 

 from the tides raised by the sun. Since E can only decrease, C must be 

 positive. The factor of proportionality, C, is the same for both since it 

 depends only upon the physical condition of the earth. 



Substituting (126) in the second of (32) we have the rate of change 

 of the day defined by 



dD C w, 



dt 2m{!t r^ 



i(--){-c^)'(ir(sy©i(-) 



The length of the month is changed by the moon's tides alone, and 

 bears a definite relation to the rate of change of the day due to the moon's 

 tides. From (32) this relation is found to be 



Zm^dP ^ 1 dP 

 D^ dt ~P^ dt 



When the sun's action on the rotation of the earth is included we have 

 therefore 



3?ni dD 



h {^<7)'m\T-=^nTm ^-^ 



D' dt Pi 

 From the formulas for the month and year we have 



\r') \S + m,-\-mJ \P') 



S-\-m^-\- m^i 



Since m^ is large compared to m^, and S is large compared to m^, this rela- 

 tion becomes with sufl&cient approximation 



(p)'=(I')'(f)' 



Then equation (128) becomes 



Zm.dD 1 j.,(mA'(P'-r>\'(P\'\dP ,.^^. 



If we integrate this equation and determine the constant of integration 

 by the present values of P and D, and if we then put P = D and solve, we 

 shall have as one of the roots the value of P at the time of the supposed 

 separation of the earth and moon. But P and D enter this equation in 

 such a complicated manner that it is not possible to express its solution 

 in finite terms. The critical values of the variables are 



P=o .=0 P-z>=o i-.(5y(^)'(;)'- 



Only the third of these critical values will arise in the applications which 

 will be made here. 



