BAKS, 111 



divided by area, viz. [ML^T^] if [M], [L\ [T] denote the units 

 of mass, length, and time. 



There are various combinations of the constants X, p which 

 are important in physics, as well as in technical mechanics. In 

 a uniform dilatation we have p 1 = p 2 = p 3 ( = p, say), l = e 2 = e s 

 (= JA), whence 



p = (X + |^)A ...................... (2) 



Hence if we write * = \ + ft, ........................ (3) 



K will denote the " volume-elasticity " or " cubical elasticity " of 

 the substance, i.e. the ratio of the uniform stress to the dilatation 

 which it involves. 



Next suppose that l = - e 2 = e, e 3 = 0, and therefore 

 Pi = p^=p, p 3 = 0, which is the case of a pure shear, 

 involving a shearing stress. According to the investigations 

 of 40, 41 the shearing stress is vr = p, and the shear is 

 rj = 2e. Hence, from (1), 



*r = /"7, ........................... (4) 



i.e. fj. denotes the ratio of the stress to the strain (appropriately 

 measured) in a pure shear. It is called the " rigidity " of the 

 substance. 



Again, suppose we have a bar stretched lengthways, but free 

 from lateral stress. We put, then, in (1), p% = 0, p 3 = 0. This 

 leads to 



pi = Ee l9 .............. . ............ (5) 



where ^ ................ 



X-f p 



This ratio of the longitudinal stress to the corresponding 

 extension is called "Young's modulus" of elasticity; its 

 technical importance is obvious. We also find 



2 = e 8 = -o- 1 , ........................ (7) 



Where 



This fraction accordingly measures the ratio of lateral con- 

 traction to longitudinal extension under the circumstances 

 supposed ; it is known as " Poisson's ratio." * 



* 8. D. Poisson (17811840). His chief contributions to acoustics relate to 

 the vibrations of membranes and plates, and to the general theory of sound- 

 waves in air. 



