164 SECULAR ACCELERATION OF THE MOON'S MEAN MOTION. [23 



It may be as well to guard against the idea that the extreme minute- 

 ness of the quantities which we have to deal with in this investigation, 

 gives rise to any uncertainty in the result. The present rate of approach 

 of the Moon to the Earth which accompanies the acceleration of its motion, 

 is less than one inch per annum, but the theory can determine this minute 

 quantity to within, say, a thousandth part of its true amount, just as 

 easily and certainly as if the quantity to be found had been any number 

 of times greater. 



I will now proceed briefly to explain the principles which I employ 

 in determining the secular acceleration, and to point out the errors which 

 vitiate the several results of MM. Plana and de Ponte"coulant which have 

 been already referred to. 



The principle of my method is simply this, viz., that the differential 

 equations must be satisfied, and that quantities which really vary must be 

 treated as variable in all the differentiations and integrations which occur 

 throughout the investigation. 



Now if e', the eccentricity of the Earth's orbit, be variable, the differ- 

 entiation or integration of any term which involves e' in its coefficient will 

 produce, in addition to the term which would result if e' were constant, 



de' 

 another term involving =- in its coefficient, supposing t to be the independent 



variable. 



In consequence of the existence of these supplementary terms, the 

 ordinary expressions for the Moon's coordinates when substituted in the 

 differential equations will not satisfy them, but will leave terms multiplied 



de' 

 by -,- outstanding. In order to destroy these terms, it is necessary to 



add terms of the same form to the usual expressions for the Moon's co- 

 ordinates. The values of these new terms may, if we please, be easily 

 found by the method of indeterminate coefficients, each of the coefficients 

 being obtained by means of a simple equation. 



If n, the Moon's mean motion, be variable, the double differentiation 

 of the Moon's coordinates will produce in the differential equations, terms 



dfL ci(*^ 



involving -y- of the same form as those already mentioned which involve -7- . 



