G. F. Becker — Current Theories of Slaty Cleavage. 5 



equi potential surfaces. Hence if crystallization takes place on 

 surfaces perpendicular to the resultant stress, these are the 

 equipotential surfaces. 



The simplest conceivable deformation is (irrotational) shear. 

 In a shear the lines of flow are rectangular hyperbolas and the 

 equipotential surfaces are rectangular hyperbolic cylinders. 

 Little more complex is the case of two shears at right angles to 

 one another. This corresponds to the axial homogeneous com- 

 pression of a cube, slab or cylinder of constant volume. In 

 discussing the mechanics of slate formation the cubical com- 

 pressibility of the mass is of small importance because after the 

 limit of elasticity is reached and flow begins, there is no further 

 change of volume. A cut (fig. 1) showing the lines of flow and 

 the plastic equipotentials is borrowed from W. J. Ibbetson's 

 well-known work on elasticity.* The circular cylinder A, B, 

 C, D is supposed compressed by uniformly distributed pressure 

 to the shorter cylinder A'' B' C D', and during the process A 

 moves to A' along the curve connecting the two, B moves to 

 B', etc. The equipotential surfaces are hyperboloids of revolu- 

 tion represented by the equation given by Ibbetson, 



2 %f — X* — 2 2 ± c* = 



where y lies in the vertical and c is a constant. They are 

 represented by full lines in the figure. 



Add to the cylinder shown in this figure a second inverted 

 cylinder at the bottom of the first, and suppose the two to 

 represent only the central portion of a slab. Then the entire 

 diagram would show the equipotential surfaces on which mica 

 scales would form if they grew at right angles to the pressure 

 in a mass subjected to pure strain. 



In mere translation, or in rotation, no work is done against 

 purely elastic or plastic resistances. Hence in a rotational strain 

 at any given instant the elastic potential is the same as it would 

 be for a pure strain of equal amplitude. There is an important 

 difference between the two cases, however, for in pure strain 

 the system of lines of flow and of equipotentials remains fixed 

 relatively to the mass, so that the motion of the particles, how- 

 ever great, is confined to the lines of flow which pass through 

 them at any instant. On the other hand, in a rotational strain 

 the lines of flow and equipotentials are not fixed relatively to 

 the mass, but only relatively to the axes oi the strain ellipsoid, 

 and, like these axes, shift continually with reference to the 

 material particles of the body undergoing strain. At any in- 

 stant, however, the equipotentials or surfaces normal to resultant 

 stress can be definitely assigned. 



* Mathematical Theory of Perfectly Elastic Solids, etc. London: Macniillan 

 and Company, 1887, p. 172. 



