192 ON FORCES OF ATTRACTION 



projection ; it is that the triangle formed by the radii vectores 

 and the line joining the two centres or fixed points, describes 

 equal solids in equal times round that line ; and the demon- 

 stration is similar to that of the first proposition, of equal 

 areas in equal times when a single force is directed to one 

 centre. It seems reasonable to conclude, that Newton had, 

 upon full consideration, found the full investigation of the 

 subject beyond the powers of the calculus as it then existed. 

 It is at least certain that, though he might have mastered it, 

 he never could have delivered his results synthetically as in 

 the Principia. 



3. The solutions on disturbing forces generally consider 

 one force as acting in the one direction, that of the radius 

 vector, and another in a line perpendicular to that radius 

 vector. Thus Clairaut {Mem. Acad. 1748, p. 435) gives these 

 equations rd*v + 2drdv = Hdx* 



rdv* d*r = Zdx*; 



r being the radius vector, v its angle with the axis, d x the 

 differential of the time, II the force to the centre, I the 

 disturbing force. So D'Alembert (Mem. Acad. 1745, p. 365) 

 takes the same course, and obtains an equation to the orbit in 

 question, depending on the integration of fTIdz, II being 

 the disturbing force acting in a line perpendicular to the 

 radius vector, and z the circular arc described with a radius 

 equal to the distance between the centre of force and the 

 vertex of the orbit. This assumes, however, that the orbit, is 

 itself nearly circular. 



4. If P = distance of E (Earth) from Moon (M)'s quadra- 

 ture, s = sin. angle of rad. vec. r with the perpendicular to a, 

 the distance of E from S, the Sun ; v = velocity of M ; then 



rdP 3P*mnsds 

 vdv = p 1 , supposing the motion of JVI to 



-t Qi 



be almost uniform. Here one of the forces acting on M 



E-f-M SxME 



is directed towards E, and is = | ; the 



JVI Jtli D JVI 



other force is in a line parallel to S E or a, and is = 



