STRAINED ELASTIC SOLIDS 405 



Treat the remaining two in a similar fashion and for temporary 

 convenience put 



. , , , g23 + 632 - 623 — 632 



a for 611 — 1, / for — r ' p for 



6 for 622 — 1, <7for r — > g for 



C ^ -LC ^'2 + ^21 , 



c for 633 — 1, Ai for — - — ' r for 

 We then have 



r' - r = ar + u + ^r' + r-n' - gr',1 



77" - V = h^ + 6V +/r + pf - rrl (7) 



r" - f ' = g^ + /V + cf ' + q^ - pv'.j 



If we take the first three terms on the right hand side of each 

 equation in (7), it is clear that they represent, as before, a differ- 

 ential displacement which at each point is normal to the corre- 

 sponding member of a family of similar quadric surfaces. As we 

 have seen, this part of the whole differential displacement still 

 leaves three certain lines orthogonal and unaffected in direction. 

 Now consider the last two terms. They represent a displace- 

 ment due to a small rotation about a line whose direction cosines 

 are proportional to p, q, r. This is readily seen by observing 

 that 



Virr)' - qn + q{p^' - r^ + r{q^' - Pv') = 

 and 



^'(rv' - qn + l(pt' - rn + ^'(q^' " pV) = 0; 



thus the small displacement of which the components are 

 '''v' — Qt'> P^' — f^', q^' — PV, is at right angles not only to the 

 line whose direction cosines are proportional to p, q, r, but also 

 to the line P'Q', whose direction cosines are of course propor- 

 tional to ^', 7/', f'. But a rotation does not disturb the angles 

 between two lines. Hence the result follows as before, so that 

 there are in every case of a small strain three particular lines. 



