Mr. HOPKINS, ON THE INTERNAL PRESSURE OF ROCK MASSES. 457 



altered by extension or compression, and it will also be twisted, so that if it were originally a 

 rectangular parallelopiped it will become an oblique-angled one, and these changes of form will be 

 indicated by the corresponding distortions of the organic remains. Now, if the directions of the 

 cleavage planes were originally determined by the state of internal tension and pressure of the mass, 

 it would seem probable that they would be perpendicular to the directions of greatest, or to those 

 of least normal pressure, or that they would coincide with the planes of greatest tangential action. 

 These hypotheses must be tested by the evidence derived from the organic forms, as will be 

 explained in the sequel, but for that purpose it will be necessary in the first place, to investigate 

 the relative positions of the lines and planes just mentioned. This investigation will form the first 

 Section of this memoir. 



SECTION I. 



Relative positions of the lines of maximum and minimum tension, and planes of 

 maximum tangential force in the interior of a continuous mass. 



1. Taking any point P of the mass, let it be made the origin of co-ordinates wyz. Let the 

 small plane s be conceived as before to pass through P, and let the forces upon it in the positions 

 specified be denoted as follows, all being referred to a unit of surface. 



(1.) When a perpendicular to the plane coincides with the axis of x, let 



iB' parallel to y, 

 z. 



(2.) When a perpendicular to the plane coincides with the axis of y, let 



!C" parallel to z, 

 A' w. 



(3.) When a perpendicular to the plane coincides with the axis of z, let 



iA" parallel to w, 



y- 



Between the six accented quantities there are three essential relations, which are easily found. 

 On the three co-ordinate axes at P, construct an indefinitely small parallelopiped whose edges 

 are ix, Sy, and Sz. The six equations of equilibrium of this element will express the conditions 

 that the sums of all the resolved parts of the forces parallel to the co-ordinate axes shaU respectively 

 be equal to zero ; and that the moments of the forces with reference to three axes, shall also 

 severally be ccjual to zero. Let us take the three latter conditions, lines through the center of 

 gravity of the element and parallel to the co-ordinate axes being taken for the axes of the com- 

 ponent couples. The tangential force parallel to the axis of x on the side Sx . S« being A', 



that on the opposite side will be — (A' + Sy) ; and the couple resulting from these forces 



about the axis parallel ta z, will be 



A'SwSzM + (A' + ~ Sy) SwSz . ^ ; 

 2 dy "' 2 



or, omitting small terms of the fourth order, 



3 N 2 



The normal force = A ; The tangential force = J , 

 When a perpendicular to the plane coincides with th( 

 The normal force = B ; The tangential force = \ 



When a perpendicular to the plane coincides with tin 



{A" 

 The normal force = C ; The tangential force = < „ 



