206 ON THE THEORETICAL VALUE OF THE [26 



It is seen, however, that these terms have very large numerical co- 

 efficients and that their sign is contrary to that of the first term, and on 

 calculation it is found that the sum of the series is less than its first 

 term nearly in the ratio of 3 to 5. 



Hence the secular acceleration will be diminished in the same ratio, 

 and its amount in a century, instead of being about 10", will be reduced 

 to nearly 6". 



No investigation of the Moon's secular acceleration can be satisfactory 

 which does not take into account terms of the nature of those which 

 give rise to the terms involving m 4 , m\ &c., above referred to. 



There is nothing to object to in the general principles of the method 

 adopted by the Astronomer Royal, but in the practical application of the 

 method I notice very grave defects. 



In the first place, the only periodic terms which are included in the 

 Astronomer Royal's expressions for T - and P - and for the factors multiplying 



a ^/<v \ *, d_ /g \ CL 

 r' dt\ r) ' dt \ 



on the right-hand side of the equations, are those which involve the 

 angle 2D or F; whereas it will be seen by a reference to my paper in 

 the Philosophical Transactions for 1853, that a great part of the co- 

 efficient of m 4 in the value of 7- there obtained arises from the combination 



n<lt 



of terms involving the angles >S', FS and F+S in the expressions for 

 the Moon's coordinates with similar terms in 



5 /\ 5, , 



8 - , or, &c. 



In the present investigation terms of the forms last mentioned are simply 

 ignored. 



In the next place, it is to be noted that, although periodic terms 



a 



depending on the angle F are introduced into the assumed values of 8 - 



and 8v, yet in Art. 12, the value of h which is the coefficient of t* in 

 the value of 8v, is found equal to Bb, quite independently of the values 

 of the coefficients e, f, g, k, and I, which occur in the terms thus introduced. 



