233-] PLANE MOTION. 



on the right 



dt dt 2r dr 2r dt dt 



The equation 



(57). 

 can therefore be integrated and we obtain 



M*-. :, (58) 



232. The two methods of integrating the differential equa- 

 tions of motion used in Art. 227 and in Art. 231 are known, 

 respectively, as the principle of areas and the principle of 

 energy (or vis viva). The former name explains itself. The 

 latter is due to the fact (to be more fully explained in kinetics) 

 that if equation (58) be multiplied by the mass of the moving 

 point, the left-hand member will represent the increase of the 

 kinetic energy of the point during the motion. 



Each of these methods of preparing the equations of motion 

 for integration consists merely in combining the equations so 

 as to obtain an exact derivative in the left-hand member of the 

 resulting equation. If by this combination the right-hand 

 member happens to vanish or to become likewise an exact 

 derivative, an integration can at once be performed. This is 

 the case in our problem. 



233. The two equations (51) and (58) can be used to find 

 the equation of the path. We have for any curvilinear motion 

 (by (4), Art. 142) 



eliminating dt by means of (51) this becomes 



