122 THE TIDAL PROBLEM. 



In the present condition of the earth-moon system we find that 



2 /p/ 2)\- /P\® 



©(^)"(fO-- 



It will appear in the computations which follow that as P and D 

 decrease in such a way as continually to satisfy (129), this function de- 

 creases rapidly until P = 0.25 and D = 0.25 approximately. For 1 ^ D > 0.25 

 it is convenient for the purpose of solving (129) to let 



<?=^^ + PJ-M (130) 



With this substitution equation (129) becomes 



dQ_ l/m,y 1 fP'-D\'/PydP (131X 



dt S\mJ Pi\P-D) VP7 dt 



If we should eliminate D from (131) by means of (130) the result would 

 have the form 



^=F{P,Q) (132) 



If we let Pq represent the present value of P, and M the total present 

 moment of momentum of the earth-moon system, then at P = Po we have 

 Q = 0, and an approximate solution of (132) is 



.P 

 Q,= / F{P,0)dP (133) 



'■■J 



Jp 



Successive approximations may be found by the series of operations 



.P 



:/ F(P,Q,)dP Q,= 



Q,= / F(P,Q,)dP Q,= F(P,Q,)dP 



■* 



Qn= I FiP,Qn-l)dP Q„+i= / F{P,Qn)dP 



./p »/p 



^0 -* 



This series of approximations approaches the true value of the integral 

 provided the upper limit does not pass beyond a point for which F(P, Q) 

 has a singularity.^ In the application of these formulas we are explicitly 

 limiting ourselves to a region in which F(P, Q) is everywhere regular. 

 The integrations can be very easily carried out by mechanical quadratures 

 to the desired degree of accuracy. 



Since we wish to trace the system back and find for what values of P 

 and D the two variables were equal, we must take P< Po in the integration. 



» Picard, Traite d' Analyse, vol. 2, pp. 301-304. 



