CHAPTER XII. 



MISCELLANEOUS METHODS AND APPLICATIONS. 



226. WE propose in this Chapter to bring together a number 

 of methods and theories relating to general classes of problems 

 which can be solved by the principles laid down in previous 

 Chapters. One of the great difficulties of our subject is the 

 integration of the differential equations of motion of a system of 

 bodies, but there are a number of cases in which all the informa- 

 tion desired can be obtained without any integration. Such cases 

 include impulsive motions, and initial motions. There are other 

 cases in which the method of integration is known. Such cases 

 include small oscillations, and problems in which the principles 

 of energy and momentum supply all the first integrals of the 

 equations of motion. We shall consider such cases here. Another 

 general problem on which much light can be thrown by the 

 theory of momentum is presented in the motion of a string or 

 chain, and this problem at the same time forms an introduction to 

 the dynamics of systems capable of continuous deformation. We 

 shall devote some space to it at the end of the Chapter. 



IMPULSES. 



227. Nature of impulsive action between bodies in 

 contact. When two bodies collide, at first their surfaces come 

 into contact at a point of each, but a little observation shows that, 

 before separation, they must be in contact over a finite area ; for 

 example, if one body is smeared over with soot, the other, after 

 separation, will show a sooty patch. It is clear therefore that 

 during the impact the bodies undergo deformation. There are 

 numberless cases in which the deformation is permanent, there 



