Sir John W. Lubbock on the Theory of the Moon. 85 



stants. In treating of the latter subject, the author shows, by an ele- 

 mentary example, the kind of operations which would be required in 

 order to determine the periodical inequalities of the moon or planets 

 by means of the variation of the elliptic constants, as proposed by 

 Poisson ; exemplifies a modification of that method arising from the 

 introduction of a certain constant quantity ; and concludes the sec- 

 tion by comparing with its previous contents the calculation of the 

 same terms according to his own methods. 



The next subject is the variation of the arbitrary constants in 

 mechanical problems, commencing with a view of the theory of that 

 variation, substantially identical with the author's paper on the 

 same subject inserted in our eleventh volume ; and followed by the 

 extension of the methods employed in that paper to a more general 

 case, as given in our twelfth volume. 



This is followed, under the head " On the Variation of the Semi- 

 axis Major of the Moon's Orbit," by the author's demonstration of 

 the theorem that the expression for that Variation contains no argu- 

 ment of long period, accompanied by a multiple of m less than m 4 . 

 As this had previously appeared at length in our pages (Third 

 Series, vol. xvii. p. 338), we need not notice it further at present. 



The concluding section, on the divergence of the numerical co- 

 efficients of certain inequalities of longitude in the lunar theory, is 

 thus introduced: — 



"The divergence of the numerical coefficients in the lunar theory, 

 made manifest by M. Plana's development of the expressions ac- 

 cording to the powers of m, presents a difficulty in a complete nu- 

 merical solution of the problem, that is, a solution intended to em- 

 brace all quantities which are sensible in practically ascertaining 

 the moon's place with the accuracy required for comparison with 

 the best observations. But the following questions naturally occur : 

 Is there any method of approximation which will serve to select the 

 more considerable terms, rejecting others ? Is the divergence due 

 chiefly to the development and expansion, according to powers of 

 m, of the divisors introduced by integration ? In the latter case the 

 difficulty might be easily avoided; but each of these questions 

 must be answered in the negative." 



In order to illustrate this point, the author has selected indiffer- 

 ently two terms in the longitude amongst those in which this di- 

 vergence is met with, and he examines their construction without 

 introducing details which do not bear immediately upon the point 

 referred to. 



After this examination he concludes the section, and with it the 

 fourth part of his work On the Theory of the Moon, with the an- 

 nexed observations : — 



" These are the most considerable, but it would evidently be im- 

 possible to employ with safety any rule of approximation which did 

 not embrace other terms. In this and in other cases in the lunar 

 theory it will be found that coefficients, when formed correctly, 

 are made up by the addition of numerous small terms, which come 

 from various sources : hence the danger of attending only to the 



