82 Notices respecting New Books. 



methods of determining the inequalities of the moon, and the peri- 

 odical inequalities of the planets, which appear to him preferable to 

 any which have yet been proposed. He observes, in the Preface to 

 the part now before us, that the expressions for the variations of the 

 elliptic constants discovered by Lagrange, were not recommended 

 by that illustrious mathematician to be employed for this determina- 

 tion, although Poisson afterwards advised that they should be used ; 

 but if his valuable life had been spared, the author observes, " it is 

 possible that his opinions upon this point would have been modified." 

 Mr. (now Sir John W.) Lubbock then proceeds to quote a letter 

 addressed to him by M. de Pontecoulant, in which the methods of 

 solving the difficult problem of the theory of the moon successively 

 employed by Clairaut, d'Alembert, Laplace, Damoiseau and Plana 

 are summarily discussed ; the remarks of M. de Pontecoulant being 

 introduced by the observation, " it will be seen that they coincide 

 with the opinions I have expressed in various places during the 

 course of this work respecting the manner in which the perturbations 

 of the moon can be computed with the greatest facility." 



" The introduction of auxiliary variables," it is next remarked, 

 " offers a wide scope to the imagination. Recently, M. Hansen has 

 proposed to arrive at the expression for the longitude by altering 

 the time in the elliptic value. It is evident that a celestial body 

 will have, at any instant, the same longitude which it has at some 

 previous or subsequent instant in the elliptic movement ; and if this 

 difference of time be calculated, the longitude can be obtained from 

 the elliptic expression by substituting for the true time some other 

 quantity. The radius vector cannot be calculated from the elliptic 

 expression by the same alteration of the time ; but M. Hansen gives 

 an expression, by means of which subsidiary terms may be obtained, 

 which, added to the radius vector calculated from the elliptic ex- 

 pression, may give its proper value." 



M. Hansen's method being applicable to all mechanical problems, 

 the author proceeds to illustrate it by a simple case, supposing 



+ x + a x = 0, 



d* 2 



and considering a a? as the disturbing function, and neglected, at first, 



"When the disturbing function is retained a further approxima- 

 tion may be obtained by one of the three following methods, ex- 

 emplified by the author : — 



1 . By substituting the value of x, a sin (t + b) in a x, and pro- 

 ceeding by the method of indeterminate coefficients. 



