STRAINED ELASTIC SOLIDS 407 



where 



ei = en^ + 621^ + e3l^ 



62 = 612^ + 622^ + e32S 



63 = ei3^ + 623^ + essS 



64 = 612613 + 622623 + 632633, 



6b = 613611 + 623621 + 633631, 



66 = 611612 + 621622 + 631632. 



(9) 



Choose for the moment a special case, letting the point Q' be 

 placed on the local axis of x' at P', so that its local coordinates 

 are ^', 0, 0. It follows from (8) that 



P"Q" = ei^" = eiP'Q' • 



Thus (61) i is the "ratio of elongation" parallel to OX', and (62)^ 

 and (63)* can be interpreted in a similar manner. It was men- 

 tioned above that two lines at right angles to each other before 

 the strain will not remain so after it. We shall show how this 

 fact is connected with the 64, 65, e^ quantities. For let us con- 

 sider Q' to be a point in the local plane of x' y' at P', its local 

 coordinates being ^' , r\ , 0. Drop perpendiculars Q'M' , Q'N' on 

 the local axes of x' and y' at P'. Let Q", M", N" be the posi- 

 tions of these points after the strain. From the result obtained 

 just above 



,2 



P"M" = ei P'M' ^, 

 P"N"^" = e2P'N'\ 



From (8) we obtain 



PW' = ei^' + 6277'2 + 2e,^'v', 

 and so 



(6162)^ 



But by the application of elementary trigonometry to the 

 parallelogram P"M"Q"N" 



P"Q" = p"M" + P"N" + 2P"M"-P"N"-co& {N"P"M"). 



