23] SECULAR ACCELERATION OF THE MOON'S MEAN MOTION. 177 



d_R 

 dv 



But ,- is a function of r and v; and since these quantities contain 



terms depending on the disturbing force and multiplied by -5- , -j-- will 



contain, in addition to the terms of the form considered by M. de Ponte- 

 coulant, other terms of the order of the square of the disturbing force, and 

 of the form 



among these there will be a term in which the angle ft + l vanishes; 

 viz., one of the form 



nl de' 



Ce dt* 



and consequently I -j dt will contain the non-periodic term - Ce'". 



M. de Pontecoulant characterises the process which I have employed at 

 the bottom of p. 402 in my Memoir (see p. 147 above), in order to find the 

 non-periodic parts of certain integrals, as " une veritable supercherie analytique." 



de' 

 Now this "supercherie" only consists in taking account of the variability of -^ , 



by putting for it the identical quantity j~ j 



M. Plana, in equation [10], p. 12, of his Memoir, finds, for the terms 

 thus objected to by M. de Pontecoulant, exactly the same values as I have 

 done, though his process entirely differs from mine. 



On this same point, in a note to p. 315, M. de Pontecoulant makes 

 the objection that in the last step of the integrations referred to I make 

 dv = ndt, contrary to the supposition I had previously employed. But my 

 object was simply to find the non-periodic parts of the integrals concerned ; 

 and it is obvious that if I had put for dv its complete value ndt <f>(v)dv, 

 where <f> (v) is a periodic function of v, this function would only introduce 

 periodic terms into the integrals, and would cause no change whatever in 

 the terms which I have found. 



But one of the most remarkable objections in the whole course of M. 

 de Pontecoulant's communication occurs in p. 316, where he says he is going 

 A. 23 



