370 Notices respecting New Boohs. 



first part forming an introduction to the second, which contains 

 accounts of the various theories in detail, although it must be 

 understood that the author adheres strictly to his intention of 

 giving an explanation of the methods, and not the actual results 

 obtained from them. After the necessary force functions have 

 been found, the ordinary simplifications introduced by neglect- 

 ing the Moon's mass, and assuming the Sun to describe an 

 elliptic orbit round the earth according to Kepler's laws, together 

 with the consequent corrections, are examined, and a numerical 

 estimate of the resulting error is given. The equations of motion 

 are next found as they are required for the methods of de Ponte- 

 coulant, Laplace, and Hill, and in addition the ten first integrals 

 arising from the equations in the problem of Three Bodies. The 

 third chapter is devoted to a discussion of undisturbed elliptic 

 motion, the expansions being made with the aid of Bessel's func- 

 tions, and the question of convergence being taken into considera- 

 tion. The two principal methods of obtaining a solution, namely, 

 by continued approximation, and the Yariation of the Arbitrary 

 Constants occurring in any orbit which may be taken as " inter- 

 mediate," can now be considered, and the equations for the 

 variations of the elements in disturbed motion are obtained in an 

 elementary way and also by Jacobi's more elegant method. In 

 this connexion some description is given of Lagrange's canonical 

 system with Hansen's extension, and some theorems of Jacobi, 

 Hamilton, and Cayley are also included. The development of the 

 forms and properties of the disturbing function brings the reader 

 to the point at which it becomes necessary to study the principal 

 methods separately. De Pontecoulant's method is very properly 

 selected by Prof. Brown as a basis for the elucidation of properties 

 common to all, and consequently receives the fullest treatment, 

 the inequalities being grouped according to their origin, and the 

 consideration of the arbitrary constants, to which a whole chapter 

 is devoted, being particularly lucid. Delaunay's method, which is. 

 next described, is important mainly on account of the high order 

 of approximation to which the literal developments are carried, 

 but also because it possesses very wide applications and possibilities 

 for development which, according to Dr. Hill, have not yet been 

 fully realized. It should be mentioned that Prof. Brown has suc- 

 ceeded in simplifying many of the explanations as they appear in 

 the Tlieorie du Mouvement de la Lune, as he has also done in 

 his account of Hansen's method. The latter has peculiar difficulties 

 and obscurities, and these the author has taken pains to remove, 

 by no means without success. 



Last of all the theories considered in detail is the one initiated by 

 Dr. Hill and based on the use of rectangular coordinates referred to 

 moving axes. It is interesting to note that these were first applied 

 to the Lunar Theory by Euler, although he had originally used 

 cylindrical coordinates for the purpose, and it is also remarkable 

 that their power in the analysis of geometrical as w r ell as dynamical 

 problems is only now becoming generally recognized. It is in 

 this most modern method of treatment of the lunar inequalities 

 that Prof. Brown's own investigations have been made. An 



