BARS 



107 



is called the "extension"; it will in general be different for 

 different directions of PQ. In the theory of elastic solids, e is 

 always a very minute fraction. We denote by e 1} e 2 , 6 8 , the 

 extensions in the directions of the principal axes. 



The ratio of the increase of volume to the original volume 

 is called the " dilatation." Denoting it by A, and considering 

 the change of volume of a cubical block whose edges are along 

 the principal axes, we have 



or A = 6! +6 2 + 6 3 , (2) 



the products of small quantities being neglected. 



There are two special types of homogeneous strain which 

 require notice. First, suppose 6! = 6 2 = 6 3 , = 6, say. Any origin- 

 ally-'^pherical portion of the sub- 

 stance then remains spherical, and 

 the extension is therefore the same 

 in all directions. The strain may 

 accordingly be described as one of 

 uniform extension; and we note 

 that = JA. 



Again, take the case of e l = 6 2 , 

 = 6, say, whilst 6 8 = 0, and therefore 

 A = 0. A square whose diagonals 

 AOC, BOD are parallel to the axes 

 1, 2 is converted into a rhombus A'B'C'D ', and since 



to the first order, the lengths of the sides are unaltered. Also 



...(3) 



tan A' OB' = i-t? = tan (J TT + e), 



1-e 



so that the angles of the rhombus are \ IT 2e. Another view 

 of this state of strain is obtained if we imagine the rhombus 

 A'B'C'D' to be moved in its own plane so that A'B' coincides 

 with AB. This is legitimate, since no displacement of the 

 body as a whole affects the question. We then see that any 

 two planes of the substance parallel to AB and the axis 3 are 

 displaced relatively to one another, without change of mutual 



