ROTATORY DISPLACEMENTS 191 



directions of lines and the positions of points in the plane. When we 

 wish to determine the rotation which a rock-mass has experienced, we 

 may then determine what rotation is necessary to bring a plane of the 

 undisturbed mass into parallelism with its displaced position, and what 

 further rotation is necessary to make a line in this plane parallel with its 

 position in the displaced plane. If a translation has also occurred, its 

 amount can easily be determined. Let us take a stratum for the plane, 

 and for the line, the intersection with it of a dike or vein. The intersec- 

 tion of this line by a second dike will determine a point. 



In figure 12 let TT be the trace of the undisturbed stratum, and B^ its 

 dip ; let tt and 8 2 be its trace and dip after displacement ; let the original 

 of line he in the first stratum be displaced into the original of bg, and 

 let the original of point / go over into the original of b' ; the problem is 

 to find the character and amount of the total displacement. The method 

 is to rotate the plane TT about its intersection with tt until the two 

 coincide, and then to rotate the plane TT about an axis perpendicular to 

 it until the originals of the lines fc and bg are parallel ; the direction of 

 the axis and the amount of the total rotation can be calculated from 

 these two partial rotations. 



As TT and tt do not intersect within the limits of the figure, we intro- 

 duce an auxiliary plane, T' T', parallel with TT, intersecting tt in the 

 original of T x. The original of ac will be parallel with the original of 

 he, a lying on the trace T' T' . In order to bring the two planes into coinci- 

 dence we must first make the original of T x horizontal; we therefore 

 rotate the whole mass about tt through the angle 8 L ,, and the former line 

 comes into the horizontal plane in T x' ; this is done as follows: g is the 

 projection of a point on the intersection of the two planes, and also on 

 the original of line bg; draw gk perpendicular to tt, and at right angles 

 to this line lay off gg", the depth of the original of g; with kg" as radius, 

 draw the arc g" g' intersecting % in g' ; g' will be the new position of the 

 original of g. T and b, being on the trace tt, are not moved by the rota- 

 tion, and therefore T' g' and bg' are the new positions of the originals of 

 T x and bg. Similarly a' e' becomes the new projection of the original 

 of ac; a goes over into a', with a' d' as the new depth of its original. So 

 far we have rotated the whole mass and have made no changes whatever 

 in the relations of its different parts. We have really merely changed 

 our plane of reference. 



Now, leaving the plane tt horizontal, we rotate the plane T T around 

 V x' until the two planes coincide. We have seen that the original of a! 

 on the plane T T lay at a depth a' d', or a d" ; and therefore, to raise this 



