Mr. HOPKINS, ON THE INTERNAL PRESSURE OF ROCK MASSES. ^61 



which by reduction gives 



// sin- S = (A - Bf cos' a cos' /3 + (^ - Cf cos' a cos' 7 + (£ - C')' cos' /3 cos' y = max ... (3). 

 Hence, if L be an arbitrary multiplier, we obtain 



\{A -5)'cos'/3 + (A - Cfcoiy + Z:| cosasina = \ 



|(^- fl)-cos'a + (fi- C)'cos'7 + L\ cos/3sin/3 = '. (^^ 



\(A - Cf cos' a+ {B- C)'cos'/3 + L] cos 7 sin 7 = 



Let us first suppose these equations satisfied by equating, in each case, their first factors to 

 zero; and for brevity put 



P = A- B, Q = A- C, R = B - C; 

 ■ ■■ P= Q- R. 



Now, substituting 1 - cos" « - cos'/3 for cos' 7, and eliminating L between the first and third, 

 and the second and third equations, we obtain 



(P2 _ Q2 _ ^2) cos2^ - 2Q2C0S2 a + Q2 = 0, 



(P2_ Q2 _ R"-) coi^ a - 2 R- cos' fi + R^ = ; 

 or since P- = (Q - R)", 



P'- Q" - R- = - 2QR; 

 .-. 2R cos' /3 + 2 Q cos' a- Q = 0, 

 SQcos'a + 2Rcos° f3-R = 0, 



which cannot hold simultaneously unless Q = R, and .-. /* = ; or A = B. This mode, therefore, 

 of satisfying equations (d) is not admissible. 

 Again, we may satisfy those equations by 



sin a = 0, cos /3 = 0, cos 7 = 0, 



a system of equations which also satisfy (2). In this case the normal to the small plane s will 



coincide with the axis of .r, i. e. with an axis of principal pressure, and therefore, these values 



ought to give the tangential force = zero, as they do. Zero is in fact a minimum value of that force. 



Similar conclusions hold with reference to the axes of y and z for the following systems of values. 



cos a = 0, sin /3 = 0, cos 7 = 0; 

 cos a = 0, cos /3 = 0, sin 7 = 0. 

 Finally, we may satisfy equations (d) by 



cos a = 0, 

 P- cos' a + R' cos" y + L = 0, 



Q' cos' a + R^ cos- /3 + Z, = 0. 

 Eliminating L, we have 



cos' /3 = cos' 7 ; 

 .-. 2 cos' /3 = 1 ; 

 .-. /3 = 7 = ± 45". 



Two other .systems of values may evidently be obtained in a similar manner, and thus equations 

 ((/) and (2) are satisfied by the three following systems of values: 



a = 90", /3 = 7 = ± IS", 1 



/3 = 90<', 7 = a = ± .1.5", (e). 



7 = 90", a = /3 = ± W". 



