2 77 .] POTENTIAL. 



(2) Show that the attraction of a circular cylinder of radius a and of 

 length /, at a point on its axis at the distance x from the nearest base, is 



(3) From the result of Ex. (2) show that the attraction of a cylinder 

 extending in one sense to infinity is = 2 TTKpa at its base. 



(4) Show that for a spherical shell of finite thickness, if the density 

 be either constant or a function of the distance from the centre only, 

 the attraction is zero at any point within the hollow of the shell, and 

 that it is the same as if the whole mass were concentrated at the centre 

 at any external point. 



(5) Show how to find the attraction of a homogeneous spherical shell 

 of finite thickness, at any point within the mass of the shell. 



(6) Show that the attraction of a solid sphere of mass M t the density 

 being any function of the distance from the centre, is = AcJ/// 2 at any 

 external point P, having the distance / from the centre, and that it is 

 directly proportional to the distance of the point P from the centre 

 when P lies within the mass. 



(7) Show that the attraction of a solid homogeneous hemisphere at a 

 point in its edge is = f K/o^V^-f 4, and that it makes with the plane 

 of the base an angle of about 32^. 



2. THE POTENTIAL. 



277. The configuration and density of any attracting masses 

 being given, the force of attraction R exerted by these masses 

 on a mass I situated at any point P can be determined both in 

 magnitude and direction. The method illustrated on some sim- 

 ple examples in the preceding articles, while theoretically quite 

 general, becomes very laborious in more complicated cases. 

 Moreover, the required resultant R, i.e. the "attraction at the 

 point P," depends as to its magnitude and direction on the 

 position of the point P ; and it is of interest to investigate its 

 variation from point to point throughout space, in a similar way 

 as was done for the example of the straight rod in Art. 268. 



Investigations of tbis kind are greatly facilitated by the aid 

 of a certain function called the potential, whose meaning and 

 use we proceed to discuss very briefly in the following articles. 



