﻿and its Relation to the Quantum Theory. 863 



The next step is to write p and s in the form 



p = a G + a v cos 2Bt + a 2 cos 4B£ + «3 cos 6B£ 4- a 4 cos 8Bt 



+ a 5 cos lOBt + a 6 cos 12B£, 

 s = & 3 cos 3B^ + ^5Cos 7 5Bt + b 7 cos 7Bt, 



where the coefficients a , a 1 , . . . a 6 , b Sj 6 5 , 6 7 , are power 

 series in w given in equation (17). Also using the 

 binomial theorem we may expand the various powers of 

 V = y/l — w cos 2 Bt as power series in w cos 2 Bt. It' the series 

 occurring in products be multiplied together, T and V 

 will consist of terms which are products of powers of sines 

 and cosines of integral multiples of Bt. By means of the 

 addition formulas the products of powers of sines and cosines 

 may be reduced to sums of linear trigonometric terms, thus 

 giving T and V as Fourier series in the time, so that 



T=T (w) + T 1 (w)cos2B* + T 2 (w) cos4B*-f . . . 

 Y = Y (w) + Vi(V)cos 2 Bt + Y 2 (w) cos 4B^ + 



It is only necessary to actually evaluate the constant term in 

 this final Fourier expansion, as the periodic terms will cancel 

 out when T and V are added together to obtain the total 

 energy, which is constant. Aiter reduction to power series 

 form in w, the constant parts T and V of the kinetic and 

 potential energies prove to be 



T = e - [l-750--053750«;--02S650w 2 --019062w 3 

 — •013920m; 4 --010658m; 5 --008473 M ; 6 -'006938^ 7 ], . (19) 

 V =- ~ [1'750- •053750w--028650w 2 --019046w 3 

 -•013884w; 4 --010613z^~-008439w; 6 --006839w 7 ], . (20) 

 while the total energy W is T + V . 



Check on Accuracy of Solution. 



One of the standard methods of checking the accuracy 

 of computations in Astronomy is to compute the energy 

 and see if it remains constant. This method could be used 

 in our problem, but would involve the calculation of the 

 coefficients of the various periodic terms in the Fourier 

 expansions of T and V, which would be extremely laborious, 

 as over twenty pages of computations are required to 

 determine the constant term alone. A much easier method 

 of checking is furnished by the fact that in motion under the 



