124 MR. LUBBOCK ON THE THEORY OF THE MOON. 



the disturbing function, the elliptic values of the moon's coordinates. The variation 

 of the disturbing function R may be obtained, according to my method, by substi- 

 tuting in the disturbing function the values of the moon's coordinates obtained by a 

 second approximation, or, as in the method of M. Poisson, by substituting in the 

 disturbing function the variations of the elliptic elements due to the disturbing force. 

 In the former case, 



d r r ' d A 'as 



In the latter, 



.„ dR. , dR. , di?v , dRy , d i? . , dR. , dR.^ 



^ being equal to /n d t. 



In the former case it is necessary to multiply 3 series by 3 series, taken two and 

 two ; in the latter it is necessary to multiply 7 series by 7 series : and the labour re- 

 quired in the one is to that required in the other about in the same proportion of 3 



to 7- The developments of ^ j— , ^ -j^ , &c., which have to be separately obtained, 



require also the same labour. 



It is important in a renewed investigation of the lunar theory, considered with a 

 view of improving the lunar tables, to obtain by some independent method the ex- 

 pressions for the coordinates given finally in terms of the mean longitude by MM. 

 Damoiseau and Plana ; for when we consider the enormous number of terms which 

 are necessary to be taken into account, and how difficult it is altogether to avoid error 

 in numerical calculations, it is hardly to be expected that their results can be entirely 

 free from error. If, however, the method of the variation of constants be adopted, 

 after the variations of the elements have been obtained, it will require no small labour 

 to effect the necessary substitutions in the elliptic expressions for the coordinates, so 

 as finally to obtain the desired comparison. The quantity of labour necessary in 

 order to bring to conclusion any solution of the problem is a very important consi- 

 deration, as every additional work, whether in algebra or numbers, brings with it 

 increased danger of mistakes, notwithstanding every care. 



The preceding remarks, however, apply particularly to the determination of those 

 inequalities which are not lowered by integration, that is, to almost all those which 



-3 



originate from the terms in R multiplied by a^ ; but with respect to others, particu- 

 larly those selected by M. Poisson, the method which he employs is very preferable. 



Laplace, in the M^canique Celeste, vol. iii. p. 17 1, alludes to an equation of long 

 period of which the argument is twice the longitude of the moon's node, plus the lon- 

 gitude of her perigee, minus three times the longitude of the sun's perigee. 



M. Poisson has shown that the coefficient of the corresponding argument in the 

 development of the disturbing function equals zero. I shall now show that this ira- 



