a ene 
1866. Adams’ Recent Astronomical Discovery. 517 
y 
But, as Adams had himself remarked, the question was a 
purely mathematical one. Other methods of investigation were 
employed ; fresh mathematicians entered the lists. Adams himself 
published two new solutions; Delaunay two more. Professor 
Donkin examined the problem by a method which had already 
been applied by Delaunay. Sir John Lubbock applied methods 
he had before employed in re-calculating other inequalities of the 
moon’s motions. Professor Cayley devised and applied a perfectly 
new method of investigation. And lastly, Plana, who, as we have 
seen, had pronounced against Adams, subjected the question to 
renewed scrutiny. The result of these investigations was to place 
the correctness of Adams’ views beyond dispute, and to assign 6” 
as the approximate value of the secular acceleration of the moon’s 
mean motion. 
We find ourselves, then, face to face again with the original 
difficulty. But before considering the solutions which appear to 
offer themselves, it may be as well to exhibit the minuteness of the 
inequality which has been the object of discussion, and one-half of 
which still remains unaccounted for. Assuming 12” as the actual 
value of the acceleration, and 6” as the value of that part for which 
theory accounts, let us consider the motion of the moon in successive 
centuries. In the first place, it is necessary to poimt out the 
incorrectness of the statement we have seen more than once made, 
that the moon moves 12” more im every century than she should 
do according to theory. That would leave the moon’s motion 
uniform. Nor, again, is it correct to say that the moon advances 
12" more in every century than she had advanced in the preceding 
century. ‘The proper statement of the nature of the acceleration 
is that the moon advances 12” farther in every century than she 
would have advanced if she had retained the same mean motion 
during the century as she had at the beginning of that interval. 
As her mean angular velocity increases uniformly, it is clear that 
her position at the end of the century is the same as she would 
have had if she had moved throughout the century with the 
velocity she had in the middle of the century, and 12” behind the 
position she would have had if she had moved with the velocity she 
has at the end of the century. In this last case she would, there- 
fore, have gained 24”; and as she actually does start with this 
velocity at the beginning of the next century, and further gains 
12” from her acceleration, her total gain is 36” during the second 
century. Similarly, it may be shown that in the third century 
she gains 60", in the fourth 84”, and so on. Thus in each century 
she moves 24" (or about one-78th part of her own breadth) farther 
than in the previous century. ‘Theory accounts only for half this 
amount, leaving to astronomers the interpretation of the other 
half, that is, of the moon’s displacement by about one-156th part of 
