165, 166] SIMPLE ROTATION. 435 



166. Rotation. Let us return to the case of the general 

 homogeneous strain given by equations 1) and let us find the condi- 

 tion that all points situated before the strain on a sphere with center 

 at the origin remain on the same sphere after the strain. 



The condition 



x '* + y'a + e '* = x * + f + 



gives 



31) fax -f a. 2 y + a 3 #) 2 -f (\x + b z y -f & 3 #) 2 



which being true for all values of x, y, s necessitates the equality of 

 the coefficients of corresponding sqares and products on both sides 

 of the equation , that is, 



V -I- If + c, 2 = 1, 

 32) aS + V + cS-*!, 



a^a^ -f &i& 2 -f 

 33) V 3 + & 2 & 3 + 



Equations 32) show that a, &, c with the same suffix are direction- 

 cosines of a line, equations 33) show that the three lines are mutually 

 perpendicular, in other words, the equations of strain are merely 

 those of transformation of coordinates, and the result of the strain 

 is merely a rotation of the body as if rigid. 



Let us obtain the analytical expression for an infinitesimal 

 rotation about an axis. Let the direction -cosines of the axis be 

 X, [i, v and the angle of rotation be do. Since we have proved in 

 57 that infinitesimal rotations may be resolved like vectors and 

 treated like angular velocities, we have the components^ of rotation, 



34) co x = ^do3, G) y = /idea, o z = vd&, 



from which by equations 119), 76, we obtain the infinitesimal 

 displacements, 



x 1 x = dx = z&y ycoz = (0fi yv) do, 



35) y 1 y = dy = x& z ZG) X = (xv gfy do, 

 # f -- 8 = dz = o x xc3 = k %i do. 



From this we obtain the substitution for the rotation considered 

 as a strain, 



28 



, 

 x = 1 x voo - y + 



36) y 1 = vdo -x+l-y- Udo 



