27] IMPULSE. IMPULSIVE FORCES. 71 



For the whole impulse we may then take as definitions 



so that the impulse is a vector quantity. We thus lose the relation 

 to the direction of the path, or of the force, in the case of a variable 

 force, but on comparing with equations 43), 13, 



dM x dM y dM z 



we have by integration 



so that the impulse of a force acting on a single particle for a 

 certain interval of time is equal to the vector increase of momentum 

 during that interval. 



The case in which the impulse of a force is of most importance 

 is that of what are known as impulsive forces, which arise where 

 actions take place between bodies in such a brief interval that the 

 bodies do not appreciably change their positions during the action, 

 although sensible changes of momenta take place. If in the equa- 

 tions above, the length of the interval ^ t decreases indefinitely, 

 while the force -components X, Y, Z increase indefinitely, the integrals 

 may still approach finite limits 



I x =lim 



In this case we can not investigate the forces in the ordinary manner 

 for the accelerations have been infinite, but the velocities and momenta 

 have received finite changes in the vanishing interval. The work 

 done is in like manner finite, though the distance moved vanish. 

 The impulse and work of all ordinary, that is finite forces acting 

 at the same time may thus be neglected, since the integral of a 

 finite integrand over a vanishing range of integration vanishes. 



On account of the third law, the action and reaction being 

 equal during the operation, the impulses of the forces on the two 

 bodies are equal and opposite, so that what one gains in momentum 

 the other loses. It is in this manner that the impact of two billiard 

 balls, or the action of a shot on a ballistic pendulum, is to be dealt 

 with. Many instruments used in electrical measurements act on this 

 principle, that the momentum suddenly communicated to a body at 



