tion R of which we have already spoken; and this function is ex- 

 panded according to terms involving cosines of the mean motions 

 of the disturbing and disturbed planet, and cosines of the difference 

 of certain multiples of these motions. This expression has been 

 treated of by various authors, and among others Mr. Lubbock has 

 himself (in memoirs read May 19 and June 9, 1831,) given the ex- 

 pansion of R in a form suited to his present object. 



The co-efficients of the terms in this expansion are arranged, as 

 usual, according to the order of the excentricities, their powers and 

 products, and to the power of the sin 2 of half the inclination. These 

 coefficients involve also certain quantities b n . where n and i have a 



variety of values ; and these quantities depend on the ratio of the 

 mean distances of the disturbing and disturbed bodies from the sun. 



Solving the differential equation which involves r, by the equating 

 of co-efficients, Mr. Lubbock finds a value for the reciprocal of r in 

 sueh terms as have been mentioned. By certain algebraical trans- 

 formations of the fractional coefficients in which i occurs, (and by 

 certain equations of condition between b 3i _ v b 3i , ^3 z --j_i > and 

 between similar quantities,) the expression for the reciprocal of r is 

 transformed and reduced, the arcs remaining as they were. 



But by the properties of the ellipse, the reciprocal of r is equal to 

 a series of terms involving the excentricities, and involving also co- 

 sines of the mean anomaly and its multiples : and hence the variation 

 of this reciprocal is equal to a similar series, involving sines and co- 

 sines of such arcs, and involving also the variations of the elliptic 

 elements. By substituting the variations of the elliptic elements given 

 by the formulae above mentioned, when we put for R its expansion, 

 we have a certain series of sines and cosines with their coefficients 

 multiplied into certain other sines of the same kind. 



It is found that the sines and cosines thus multiplied produce, by 

 trigonometrical transformations, arcs identical with those which were 

 found in the value of the reciprocal of r obtained by the former 

 method ; and the coefficients are also found to be identical with those 

 resulting from the former transformations and reductions. 



We have not thought it necessary to verify the somewhat complex 

 reductions by which Mr. Lubbock has shown the identity of the results 

 obtained by these two methods. The mode of proceeding is per- 

 fectly satisfactory, and the truth of the conclusion might have been 

 foreseen. The reductions, however, by which identity was to be 

 exhibited were by no means obvious : and we conceive it not un- 

 likely that the development of them may sometimes be of use in 

 enabling us to judge which of the two methods of solution may be 

 applied with most convenience in particular cases. 



We are of opinion that this Paper is well worthy of being printed 

 in our Transactions. (Signed) W. Whewell. 



Geo. Peacock, 

 h. coddington. 



