76 STATICS OF SYSTEMS OF PARTICLES 



Also the denominator sin a sin (a d6) is the increase in 

 sin a when a changes from a dO to a, and this in the same way 



is equal to 



d (sin a) a a 



* - - du or cos a du. 

 da 



Thus the original fraction is equal to 



dT 



Te^ dT 



- or sec a - - 

 cos a dd du 



T dT 



Thus - = sec a r> 



sin a du 



fJT 

 or T g = tan a (12) 



When in the limit the particle PQ is supposed to become van- 

 ishingly small, T p and T Q become indistinguishable. Let us denote 

 either by the single letter T, so that T is now simply the tension 

 at a point at which the normal makes an angle 6 with that at A. 

 If the string is just on the point of slipping in the direction APQ, 

 the angle between the reaction R and the normal of either Q or P 

 will be e, the angle of friction. Thus we shall have 



7T 



--<, 



so that tan a = cot e, and equation (12) becomes 



54. If the contact between the surface and the string is per- 



dT 

 fectly smooth, e = 0, so that = 0. It follows that T is a con- 



CLv 



stant ; i.e. the tension is the same at all points of the string. Thus 

 the tension of a string is not altered by its passing over a smooth 

 surface, the result already given in 36. 



