STRINGS 



75 



Let A be the point of the surface at which the string leaves it. 

 Let the normals to the surface be drawn at A, P, Q, and let the 

 normal at P make an angle 6 with the normal at A. If the points 

 A, P, Q come in this order, as in fig. 40, the normal at Q will make 

 with the normal at A an angle slightly greater than 6, say 6 -f- dd, 

 so that dd is the small angle between the normals at P and Q. 



FIG. 40 



With this notation the angle between the tensions T p and T Q is 

 TT dd. Let the angle between the reaction R and the tension T p 

 be a, then the angle between the tension T Q and H is TT a + dd. 

 Thus we have 



R 



sin(7r dd} sin(7r a + dO) 

 Since sin(?r a + dd) = sin (a dO), we have 



T 



sin a 



T 



sin a 



sin (a dd) 

 and by a known theorem in algebra, each fraction is equal to 



sin a sin (a dd) 



Now T Q T P is the increase in T when 6 changes from 6 to 6 + dO, 

 and this, in the notation of the differential calculus, may be written 



