Ivii 



a system. The theorem achiiits of being proved by conside- 

 rations more elementary, but was suggested to the author 

 by the analysis above described ; which may be extended, by 

 continuing the developments (15), (16), to the case of one 

 planet disturbed by another, and to a more accurate theory of 

 a satellite. 



Without entering into any farther account at present of 

 the attempts which he has made to apply the processes and 

 notation of his calculus of quaternions, or method of vectors, 

 to questions of physical astronomy, the author wished to state 

 that he had found those processes, and that notation, adapt 

 themselves with remarkable facility to questions and results 

 respecting Poinsot's Theory of Mechanical Couples. A single 

 force, of the ordinary kind, is naturally represented by a vec- 

 tor, because it is constructed or represented, in mathematical 

 reasoning, by a straight line having direction ; but also a 

 couple, of the kind considered by Poinsot, is found, in Sir 

 William Hamilton's analysis, to admit of being regarded as 

 the vector part of the product of two vectors, namely, of those 

 which represent respectively one of the two forces of the cou- 

 ple, and the straight line drawn to any point of its line of 

 direction from any point of the line of direction of the other 

 force. Composition of couples corresponds to addition of such 

 vector parts ; and the laws of equilibrium of several forces, 

 applied to various points of a solid body, are thus included in 

 the two equations, 



2)3 = 0; 2(a/3 - i3a) = ; (18) 



the vector of the point of application being a, and the vector 

 representing the force applied at that point being j3. The 

 condition of the existence of a single resultant is expressed by 

 the formula, 



Si3.S(a/3 - i3a) + S(ai3 - ^a) . 2/3 =: 0. (19) 



Instead of the two equations of equilibrium (18), we may 

 employ the single formula 



S . o/3 - - c, (20) 



