lvii 
a system. ‘The theorem admits 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 caleulus 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, 
=B=0; (aS — Ba) = 0; (18) 
the vector of the point of application being a, and the vector 
representing the force applied at that point being B. The 
condition of the existence of a single resultant is expressed by 
the formula, 
SPB. 3(aB — Ba) + (aS — Pa).-SB=O0. (19) 
Instead of the two equations of equilibrium (18), we may 
employ the single formula 
=.a8 = =, (20) 
