21] ON THE SECULAR VARIATION OF THE MOON'S MEAN MOTION. 157 



Now, if the eccentricity of the Earth's orbit be supposed to remain 

 constant, this is the true form of the expressions for the Moon's coordinates ; 

 but if that eccentricity be variable, the author shews that the differential 

 equation cannot be satisfied without adding to the expression for the re- 

 ciprocal of the radius vector, a series of small supplementary terms depending 

 on the sines of the angles whose cosines are already involved in it, and 

 to the expression for the longitude, a series of similar terms depending on 

 the cosines of the same angles ; all the coefficients of these new terms 

 containing as a factor the differential coefficient of the eccentricity of the 

 Earth's orbit taken with respect to the time. 



The author first determines as many of these terms as are necessary 

 in the order of approximation to which he restricts himself, and then takes 

 them into account in the investigation of the secular acceleration. The 

 expression which he thus obtains for the first two terms of this accele- 

 ration, is, 



_/3, 

 U 



According to Plana, the corresponding expression is 



f3 



It will be observed that the coefficient of the second term has been com- 

 pletely altered in consequence of the introduction of the new terms. 



The numerical effect of this alteration is to dimmish by l"'66 the 

 coefficient of the square of the time in the expression for the secular 

 acceleration ; the time being, as usual, expressed in centuries. 



It will, of course, be necessary to carry the approximation much further, 

 in order to obtain such a value of this coefficient as may be employed 

 with confidence in the calculation of ancient eclipses. 



