Forces on the Motion of the Moon. 531 



tions now applied, it is extremely unlikely that further errors 

 showing a periodicity of this kind are present. 



The ordinary gravitational theory must at the outset be 

 excluded. It has been so thoroughly investigated that there 

 seems to be no likelihood of a term with so large a coefficient 

 as 12* having escaped detection. The term has frequently 

 been ascribed to planetary action, but this action has now been 

 treated in detail by several writers, and, in my own work on 

 the subject, several thousands of terms were examined. Not 

 the slightest possibility of the existence of a coefficient of any- 

 where near this magnitude has arisen. It may be noted that 

 we are searching for a term nearly as great as the largest known 

 term produced by planetary action, and further, that the next 

 largest term has a coefficient only one-tenth of the required 

 size. To ascribe the term to this cause would mean a defect 

 in the theory so great that a total reconstruction would prob- 

 ably be necessary if an omission had been made. 



i. Method of investigatio?i. — The chief point is the period 

 of the term, which is known from observation to be about 270 

 years. Now from the lunar theory we know that this period 

 can be produced in three ways. It may be the effect of a 

 nearly constant force on a term having this period, already 

 present in the expressions for the moon's longitude. If the 

 force be periodic, the period of the force must be either the 

 period of the effect, or else the combination of the period of 

 the force with one of the known periods in the moon's motion 

 must produce the period of the effect. Now the period of the 

 effect being 270 years, we must either look for a force having 

 a period of 270 years, or else we must look for a force whose 

 period differs from one of the lunar periods by a small quan- 

 tity which runs through its changes in 270 years. The hypoth- 

 esis of a nearly constant force is extremely improbable. 



There is another point in the investigation that makes it 

 easy to reject some of the hypotheses which may be brought 

 forward. The mean annual motion of the perigee, as derived 

 from theory and observation, shows no outstanding difference 

 greater than 3/10 of a second ; and a similar statement may be 

 made with respect to the node. If, therefore, we assume the 

 existence of some force which gives a motion to the perigee, 

 or to the node, much greater than this amount, the hypothesis 

 must be rejected. 



5. In the examination of long-period terms in the moon's 

 motion there is one point of great importance in the consider- 

 ation of the magnitude of the disturbing force. If the force 

 has a period near that of the month, this period not depending 

 on the mean motion of the moon, the factor which is chiefly 

 responsible for the large coefficient depends on the square of 



