26] ACCELERATION OF THE MOON'S MEAN MOTION. 207 



The result of this is to reduce the secular acceleration practically to 

 its first term only; which accounts for the coincidence of the Astronomer 

 Royal's value with that of Laplace. 



It may also be remarked in reference to Art. 11, that although terms 

 involving the argument 2F or 4Z) may be properly omitted, we must put 



and cos'F - + - cos 2F, 



and the constant terms in these latter quantities should be taken into 

 account. 



After these general remarks, we will enter a little more closely on the 

 consideration of one or two points in the investigation which are important. 



Adopting the Astronomer Royal's notation, let 

 a denote the Sun's mass, 



A the semiaxis major of the Sun's (or Earth's) orbit, 

 E the eccentricity of the orbit, 

 R the radius vector at any time. 



Then it may be shewn, as in the paper before us, that the mean value of 

 o~.tr 1 cr / 3 jiA 



2? 1S A* (T^j* ' = * 1 + 2 E ) nearly> 



Hence if E receive the variation 8E in the time t, this quantity will be 

 increased in the ratio of 1 + 3ESE to 1 nearly, or in the ratio of 1+bt 

 to 1, calling 



Having arrived at this point, the Astronomer Royal assumes that the 

 variation of the disturbing forces due to the variation 8E in the eccentricity 

 of the Sun's orbit will be represented by supposing 



T to be replaced by T(l + U), 

 and similarly P to be replaced by P(l+bt), 



