OF THE MOTION OF A SYSTEM. 201 



If the forces P, Q, i2, are the results of attractions 

 directed to fixed points, and of attractions of the liodies 

 to each other, the quantity 2m (Pdar+ Qdy + l?dz) is an 

 exact fluxion. For the part which depends on the attrac- 

 tion to fixed points is an exact fluxion, because the forces 

 in the three directions are obtained by the resolution of 

 single forces acting in given lines, each of which must 

 afford a true or exact variation when resolved, so that their 

 sum, however combined, must still be an exact variation. 

 And with respect to the parts depending on the mutual 

 attractions of the bodies of the system, if we call thQ dis- 

 tance of m from m\ f, and the attraction of m for m, m'F, 

 the part of m (Pdo: + Qdy + jRdz) that relates to this 

 attraction will be mudF^'f^ the fluxion dy relating to the 

 change of the coordinates of m only ; but since reaction 

 is alway equal and contrary to action, the part of rd {P'dx' 

 -f Q'dy' + Rdz') depending on the action of m or m' is 

 equal to — mm'Fdy, supposing d'fio relate to the change 

 of the coordinates of m' only: consequently the whole 

 effect of the reciprocal action of m and m' is represented 

 by the product — mm'Fdf, df being the total variation of 

 /; and Pd/is an exact fluxion whenever P is a function of 

 /, or when the attraction is dependent on the distance, as 

 we suppose to be the case with respect to attractive forces 

 in general. Consequently the sum of all such actions 

 must be expressed by an exact fluxion, whenever the 

 forces concerned depend on the attraction of the bodies 

 of the system for each other, or for any fixed points. If 

 then we suppose this fluxion to be d^, and if we call the 

 velocity of m, v, that of m\ v', . . . , we shall have 



2wit;2=:c-f2^ (R) 



