210 NEWTON S PRINCIPIA. 



and finally -stops after a time given by the first equation, 

 having described a space given by the second. It is ma 

 nifest that the particle remains at rest, until it is disturbed 

 by some new force. 



But here we have a remarkable singularity in the equa 

 tions ; for according to them, as t increases v l ~ n does not 

 remain equal to zero, but becomes negative. What is the 

 explanation of this ? It must be sought for in the nature 

 of a differential equation. There are always two species 

 of integrals. One called the &quot;general integral,&quot; which 

 contains the full number of arbitrary constants, and ano 

 ther, called the &quot; singular solution,&quot; not included in the 

 former, and which does not contain the full number of ar 

 bitrary constants. These latter in dynamical problems are 

 usually of little value, because they do not agree with the 

 initial conditions of motion. But, if by any chance they 

 should satisfy these conditions, it is possible that they may 

 be the true representatives of the subsequent motion. The 

 choice between them and the general integral must be 

 founded on extrinsic considerations. The differential equa 

 tion we started with, is a mere statement of the forces, and 

 must be true throughout the motion. This motion must 

 therefore be represented either by the general or the sin 

 gular solution. We have seen that the general solution 

 only represents the motion up to a certain time ; after that 

 we must have recourse to the singular solution. If we 

 proceed to find this, by the usual methods, we arrive at 

 the solution 



v=0, 



which we see represents the motion subsequently to the 

 above mentioned time.* 



One of the most remarkable facts connected with motion 

 in a resisting medium is the existence of a &quot; terminal ve- 



* Duhamel, Cours de Mecanique. 



