274-] 



ATTRACTION. 



equal attractions are produced at P by the elements dS at H and dS' at 

 //' cut out by any thin cone whose vertex is the inverse point P'. We 

 have, as in Art. 272, for the mass elements at H and H 1 cut out by the 



COIle l J 



and for the corresponding attractions at 



but these expressions are equal since /"77C = P'ff'C and the 

 similar triangles give r/PH= a/p, r'/PH' = a /p. 



As, moreover, these attractions make equal angles with CP t their 

 projections on this line are equal, and their resultant is 







To form the final resultant, this expression must be integrated over 

 the whole sphere, and as the summation of the double cone gives 

 I //a = 2 TT, we find 



D 

 R= 



where M denotes the whole mass of the shell. Hence, the attraction 

 of a homogeneous shell on an external point is the same as if the whole 

 mass of the shell were concentrated at the centre of the shell. 



274. (c) Attraction at the surface. If the point P approaches the 

 surface from within, the attraction remains constantly zero ; if P 

 approaches the surface from without, the at- 

 traction KM/p 2 approaches the limit nM/a*. 

 For a point on the surface the attraction is 

 the arithmetic mean of these two values, viz. 



R= 27TKp. (l4) 



This can be shown as follows (Fig. 83). 

 The element of mass at H is 



pdS=p- rWa>/cos PHC ; 



Fig. 83. 



its attraction at P is Kp dw/cos PHC, and as the angles at P and H are 

 equal, the projection of the attraction on PC = Kprfv. For a point 

 on the surface \dw= 271-. Hence the total attraction at /Ms = 27r/c/o. 



