125-129] EXAMPLE OF STRAIN. 121 



state. We suppose As , having one and the same particle always 

 at one of its ends, to be indefinitely reduced. Then we define the 



^V o .... ^V o 



limit of the ratio , when A is indefinitely diminished, to 



L^SQ 



be the extension of the string at the point to which As approaches. 

 Thus the extension is measured by the ratio of the increase of 

 length to the natural length. 



In an extensible string the tension at any point is a constant 

 multiple of the extension at the same point, or, if X is a certain 

 constant depending on the material of the string, and of the 

 dimensions of force, the tension is Xe, where e is the extension. 



The constant \ of an extensible string is the modulus of 

 elasticity of the string. 



When the mass of the string is neglected the tension is 

 constant throughout, and so is the extension*. In this case the 

 extension is equal to (I - Z )/J , where 1 is the length of the string 

 in its natural state (called the natural length), and I is the length 

 of the string when extended. 



128. Inextensible string. A string which exerts tension, 

 but is never sensibly extended, must be thought of as the limit to 

 which an extensible string approximates, when the extension e 

 remains always less than a number which we agree to neglect, 

 but the modulus X is very great, in such a way that the product 

 Xe is of the order which we agree to retain. 



129. Spring. The word spring is used in Rational Mechanics 

 to express an idea suggested by generalising the idea of an 

 extensible string. We imagine a line of particles in which the 

 stress across a plane normal to the line is always either pressure 

 or tension, and a natural state of the system in which it vanishes. 

 When the length of any element exceeds its natural length there 

 is extension, and tension equal to the product of X and the 

 extension. When the length of any element is less than its 

 natural length there is contraction, measured by the ratio of the 

 diminution of length to the natural length, and pressure equal to 

 the product of X and the contraction. The constant X is the 

 modulus of elasticity of the spring. 



* There is a reservation for contact with a rough body. 



