RECIPROCAL ACTION OF TWO SHELLS. 



333 



If 6 and 6' are the angles which the elements ds and ds' make 

 respectively with the right line OM which joins them, and e the angle 

 of these two elements, we have 



- = cos<9', 



and we get 



r = I cos - - cos cos 0' I ds 



2 



If we consider the action of ds upon ds' and take the distance 

 r as positive in the direction MO, we must change the sign of the 

 force and replace the angles and 0' by TT - 6 and TT - B' t which 

 does not change the sign of the product of the cosines. 



Let us represent by d^ the action of ds upon ds' , which is an 

 infinitely small quantity of the second order, and consider this force 

 to be repulsive ; we shall have finally 



351. We may give another form to this expression, which is more 

 convenient for estimating the work. 



Fig. 83. 



Let C and C' (Fig. 83) be the edges of two shells, ds and ds' 

 the elements at P and P', and let us count the arcs s and s' respec- 

 tively from the fixed points O and O'. 



