120 SYSTEMS OF FORCES. [CHAP. VII. 



A case of great importance is that in which p is regarded as 

 infinitesimal. For small values of p there is clearly an order of 

 approximation to which it is correct to write the above equation 

 cJT 

 ds = * ^ anC * p ^ k e ^ n sma ll er than a number which we agree to 



neglect ; then, to the same order of approximation, T is constant. 



When the mass of a string is neglected the tension is constant 

 throughout any portion which is subject only to bodily forces and 

 terminal tensions. 



A string of which the mass is neglected will frequently be called 

 a thread, and one of which the mass is not neglected a chain. 



126. Reaction of string in contact with surface. When 

 the string is in contact with a body, there will be forces which are 

 not bodily forces arising from the mutual actions of the particles 

 of the string and the particles of the body exerted across the 

 surface of the body. When the surface is smooth the resultant 

 of these actions upon any element of the string will be directed 

 along the normal to the surface, and will therefore have no 

 resolved part in the direction of the tangent line to the string. 

 The equation (A) will therefore still hold, and, as before, it will 

 follow that when the mass of a string is neglected the tension is 

 constant throughout any portion in contact with a smooth surface. 



Whether the surface is smooth or not there will be a finite 

 pressure per unit length at any point. Taking as before P, P to 

 be the positions of two particles of the string near together, the 

 reaction of the surface upon the part of the string between P and 

 P can be resolved into a pressure along the normal at P, and 

 tangential forces in the tangent plane at P to the surface. If PP 

 is of length As, then these will in general be the products of 

 finite quantities and As. Thus if JSAs is the pressure on the 

 element, R is the pressure at P per unit length, and R is always a 

 finite quantity. 



127. Extensible string. When the string changes its 

 length there is always a particular length for each portion of it 

 such that the tension is throughout zero. When every portion 

 has this length the string is said to be in its natural state. 



Let As be the length of a short piece of the string in its 

 natural state, and As the length of the same piece in any other 



