20 G. F. BECKER FINITE STRAIN IN ROCKS. 



From these two equations v can at once be eliminated, since tan v cot v = 1. 

 Writing out this equation and reducing, one rinds— 



ton2 ,«- 2— ^ 1 + e) + a(1+ '> - 



P~a !, + (1+/)*— (1 + e) 1 



The equations for v and p. can be combined to simpler forms. It will 

 be found on trial that the values already deduced lead to — 



tan + A0 = ( i + , )_ ( i +/) 5 tan (v - /,) = (1 + ^ (1 + f) - (6) 



The angle v — ;>. is the angle of rotation, so that the condition of no 

 rotation is evidently a = b. When the strain is infinitesimal, a — b is 

 infinitesimal, while 1 + e + 1 + ,f approaches 2. Hence v — fi is zero 

 for vanishing strain. If the common limiting value of v and ,u is v a , tan 

 (y + , a ) = tan 2 v , or — 



a 4- b 



tan 2 v_ = 



(1 + -(!+/)■ 



Of course this same value is obtained by letting a, b, e and / approach 

 zero in the formulas for tan 2 // and tan 2 v. Thus v — v = v o — //. It 

 is evident that as rotation proceeds new fibers of matter constantly suc- 

 ceed one another in the position of axis, the whole series of fibers in the 



unstrained mass forming a wedge, v — ,a or — — — • 



Lines of constant Direction. — Lines parallel to o z retain their direction 

 relatively to the x y plane throughout strain. If the mass were inflexible 

 and subjected to rotation, only these lines would maintain their direc- 

 tion ; but when there is strain two other lines may retain their original 

 direction, the two coinciding in the limiting case which separates that 

 of three such lines from that of one. 



If /. is the angle which any line in the x y plane makes with o x before 

 strain and / the angle which it makes after strain, then — 



tanx=y- J = a r (1+ { )tanx - 



x 1 -f- e + b tan /. 



If X = /. this gives — 



which represents two real lines, unless the quantity under the radical 



