87-4 HISTORY OF MECHANICS. 



slants may most conveniently be reserved for secular inequalities and 

 inequalities of long periods. 



MM. Lubbock and Pontecoulant have made the mode of treating the 

 Lunar TheQry and the Planetary Theory agree with each other, 

 instead of following two different paths in the calculation of the two 

 problems, which had previously been done. 



Prof. Hansen, also, in his Fundamenta Nova Investiyationis Orbiics 

 verce quam Luna perlustrat (Gothce, 1838), gives a general method, 

 including the Lunar Theory and the Planetary Theory as two special 

 cases. To this is annexed a solution of the Problem of Four Bodies. 



I am here speaking of the Lunar and Planetary Theories as Mechan- 

 ical Problems only. Connected with this subject, I will not omit to 

 notice a very general and beautiful method of solving problems 

 respecting the motion of systems mutually attracting bodies, given by 

 SirW. K. Hamilton, in the Philosophical Transactions for 1834-5 

 ("On a General Method in Dynamics"). His method consists in 

 investigating the Principal Function of the co-ordinates of the bodies: 

 this function being one, by the differentiation of which, the co-ordinates 

 of the bodies of the system may be found. Moreover, an approximate 

 value of this function being obtained, the same formulae supply a means 

 of successive approximation without limit.] 



9. Precession. Motion of Rigid Bodies. The series of investiga- 

 tions of which I have spoken, extensive and complex as it is, treats the 

 moving bodies as points only, and takes no account of any peculiarity 

 of their form or motion of their parts. The investigation of the motion 

 of a body of any magnitude and form, is another branch of analytical 

 mechanics, which well deserves notice. Like the former branch, it 

 mainly owed its cultivation to the problems suggested by the solar 

 system. Newton, as we have seen, endeavored to calculate the effect 

 of the attraction of the sun and moon in producing the precession oj 

 the equinoxes ; but in doing this he made some mistakes. In 1747, 

 D'Alembert solved this problem by the aid of his " Principle ;" and it 

 was not difficult for him to show, as he did in his 0^>uscules^ in 1761, 

 that the same method enabled him to determine the motion of a body 

 of any figure acted upon by any forces. But, as the reader will have 

 observed in the course of this narrative, the great mathematicians of 

 this period were always nearly abreast of each other in their advances. 

 Euler, 10 in the mean time, had published, in 1751, a solution of the 



Ac. Berl. 1745, 1750. 



