232 NOTE ON THE INEQUALITY IN THE MOON'S LATITUDE, ETC. [29 



At the Meeting of the Society in March last, the Astronomer Royal 

 gave an investigation of the inequality in latitude based upon the equations 

 supplied by the "Factorial Tables" of his "Numerical Lunar Theory." About 

 one portion of this investigation I wish to make a remark which seems 

 to be important. 



The Astronomer Royal forms his equations with reference to the fixed 

 ecliptic, and, by integrating them, derives the value of the disturbed latitude 

 above the fixed ecliptic, whence the latitude above the variable ecliptic is 

 immediately deduced. 



The latitude so found contains not only the inequality in latitude 

 required, but also the small residual terms 



Bt {'003 sin nt-C + '005 sin nt-2Nt + C\}, 



which the Astronomer Royal rejects, attributing them to accidental errors 

 in the last places of the decimals employed. 



I shall presently attempt to shew that these terms must indeed be 



rejected, though not for the reason here supposed, but because they are 



destroyed by other terms which would be found by a more complete in- 

 vestigation. 



It should be remarked that if terms of the above form really existed, 

 they would, notwithstanding the smallness of their numerical coefficients, 

 ultimately become much more important than the other terms in which t 

 does not occur in the coefficients. 



I propose to prove that in the complete solution of the differential 

 equations no terms of the above-mentioned form can occur, supposing the 

 displacements of the plane of the ecliptic to be proportional to the first 

 power of t. The method which I employ for this purpose is the following. 



Instead of solving the differential equations of motion with reference 

 to the fixed ecliptic and then transforming the results so as to make them 

 apply to the variable ecliptic, I first transform the differential equations 

 of motion, so as to make them refer to the variable ecliptic, and when 

 this is done, it is found that the terms which contain t in their coefficients 

 disappear completely from the differential equations, so that the solution 

 may be effected by the ordinary methods without any difficulty. 



Employing the same data and notation as the Astronomer Royal, and 

 taking into account only the terms which are independent of the Moon's 



