XXXVll 



of an orbit, and the equable description of areas, under one 

 common formula, namely, the following : 



dp dp 



PdJ-dT'^^^^"^*' 



from the general expression of a central force, namely, from 

 the equation 



dV_dV 



Pa^2—-a72P = ^' 



which asserts merely the coaxality of the vector p and the 



force -r^, or the existence of one common line along which 



this vector and this force are (similarly or oppositely) directed. 

 Since the suggestion above acknowledged was made. Sir 

 William Hamilton has proposed to himself to express by an 

 equation, on the principles of the method of vectors, the pro- 

 blem of any number of bodies attracting according to New- 

 ton's law : and has arrived at the formula 



d^a _ m + Am 



d? "■ - Aa V(-Aa^)' ^""^ 



which may also be thus written, 



d^ _ ^ '>n^ f . 



de - {a-a') ^/{-(a- a'f\ ' ^^^ 



and from which he has deduced anew the known laws of the 



centre of gravity, of areas, and of the vis viva, under the 



forms : 



dV 



Jt) S.^;^a = 0; (c) 



d „ / da da \ - 



d^^-"\v«Trd7"J=^' ^°> 



d ^ m fdaV d „ mm' _ , . 



a is the vector and m the mass of one body ; a' and m' of another; 



