Mk. HOPKINS, ON THE INTERNAL PRESSURE OF ROCK MASSES. 465 



tile- second of tlie systems (i ), (Art. 4), and the corresponding value of the tangential force is 



r, = i ( J - C), 



the absolute maxiniuni value of the tangential force acting on a plane of indefinitely small and 

 constant area (s), passing through any assigned point P of the solid body, and capable of assuming 

 any angular position about that point. For this maximum value, the normal to the small plane, 

 maylie in two positions, both in the plane of wx, and bisecting the angleb etween the axes of 

 ,r and z, the one above and the other below the axis of .i', those axes being so taken as to coincide, 

 the former with the direction of the greatest principal tension at P, and tiie latter with that of the 

 least. If one of the principal tensions be changed into a pressure, it must be regarded as a 

 tiegative tetisiun, and therefore as the least principal tension, and its direction taken as the 

 axis of X. In this case we shall have T2 = \{A + C). If there be two pressures, the greatest 

 will = — a. If all the principal forces be pressures, the least pressure will be - A, and the 

 greatest pressure - C, and therefoi-e ^2 = ^ (C - A). Thus T^ will in all cases be the algebraical 

 difference of the greatest and least principal tensions, considering pressures as negative tensions.* 



G. .\s an elucidation of the subject, I shall consider a few particular cases. 



(1) Suppose B = C ; the normal tensions will be the same for all positions of the plane s, in 

 which its normal lies in the plane yz, and there will be an infinite number of positions of the 

 plane s corresponding to the maximum tangential action, such that the locus to the normals of * 

 will be a conical surface whose axis is that of x, the semi-vertical angle of the cone being 45". 



(2) If the mass at any proposed point (P) be acted upon only by two tensions acting as principal 

 ten.sions, these must be considered as the axes of * and y, the axis of z, that of least principal 

 tension (supposed here = zero) being perpendicular to the plane of the two tensions. 



(.3) If there be only two principal tensions, as in the last case, but one of them become a 

 pressure, the direction of this latter must be taken as the axis of z, that of least tension. 



(4) If both these principal tensions become pressures, the line perpendicular to the plane in 

 which they act, must be taken for the axis of x, (the axis of greatest principal tension), and the 

 direction of the greatest pressure for the axis o{ z. 



The axes of x and z, those of greatest and least principal tensions being known, the two positions 

 of the plane of maximum tangential action are immediately known. 



(j) Let PQRS represent a plane section of an elementary parallelopiped of the body parallel 

 to two opposite sides, and suppose PQRS a square. Let the forces on the 

 element be entirely tangential and parallel to the plane of the paper, there 

 being no force perpendicular to that plane Then (Art. 1) the tangential 

 force on each side of the element will be the same; let it = f and act on each 

 side in the directions inilicated by the arrows. Also let Pr/rS be the section 

 of the same clement, sujiposing the forces / not to act; then it is manifest tliat 

 tliese forces produce an evtension = SQ — Sq in the direction SQ, and a com- 

 presnion = Pr - PR in the direction RP perpendicular to SQ. In fact the 

 forces/ may be resolved into/, cos 4.')" ])arallel to SQ, extending each jiarticle in that direction, 

 and an equal force compressing the ])articles perpendicular to SQ. Tiie former will act as a 

 principal tcn.sion, the latter as a principal pressure. If A and - C be their values referred to a 

 unit of surface, we must iiave 



A . QR sin 4.V' =/. QR cos 4.0", 



.-..kI C . QR cos 45" =/ . QR sill 1.V' ; 



.-. A=f, and C=/; 



Since T^ = {A - €)"*. Tg - ±(yl - C). All tiotitc yyi llit* in-giiiive sign !■ oniiitwl in the tc-itt. n* nltogcther unessential. 



