408 

 Hence 



RICE 



cos {N"P"M") = 



66 



(6162)" 



AET. K 



(10) 



and similar results can be obtained for the other pairs of axes. 

 A glance at Fig. 4 shows that the rectangle P'M'Q'N' has 

 suffered a shear to the shape P"M"Q"N". (It is in general 

 also subject to a rotation.) The shear is measured by the angle 

 L"P"N" whose sine is by equation (10) equal to 66/(^162)^ If 

 the strains are sufficiently small we can simplify this. Recalhng 

 the original definitions of the era coefficients in (4), we see that 



^11 — 1, 622 — 1, 633 — 1, 623, 632, 631, ei3. 612, 621 



X' 



Fig. 4 



are small compared to unity if the relative displacement of two 

 points is a small fraction of their distance apart. Hence, by 

 (9), ei, 62, ez each differ from unity by a small amount. Also in 

 the definition of ee the third term is the product of two small 

 quantities, the second term differs from 621 by a small fraction 

 of 621, and the first term differs from e^ by a small fraction of 612. 

 Thus, apart from a neghgible error, the sine of L"P"N" is equal 

 to 612 + 621. The angle being also small in this case, its value, 

 that is the shear of the lines originally parallel to OX' and OY', 

 is practically 612 + 621," this in fact measures very closely the 

 amount by which the angle between these lines has changed 



