132 NOTES TO BOOK VI. 



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 Mechanical 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 of mutually attracting bodies given by 

 Sir W. R. Hamilton, in the Philosophical Transactions for 

 J 8345, (" 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 diiferentiation 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. 



I may mention here that a new abstract term, intro- 

 duced because a general mechanical principle can be 

 expressed by means of it, has lately been much employed 

 by the mathematical engineers of France, MM. Ponce- 

 let, Navier, Morin, &c. The abstract term is Travail, 

 which has been translated labouring force ; and the prin- 

 ciple which gives it its value, and makes it useful in the 

 solution of problems, is this ; that the quantity of 

 labouring force which will overcome a given resistance, or 

 produce a given effect, is the same by whatever mechani- 

 cal contrivances the force be applied. This principle is 

 mathematically equivalent to the conservation of vis viva. 

 See Poncelet's Mecanique Industrielle, &c. 



