464 



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



value, since maxima and minima occur alternately. To ascertain whether any value of 6 between 



dr 

 and 90° does render — = 0, we must see whether such value can be derived in anv of the 

 (19 



three cases above, by equating to zero the expressions within the brackets in equations (/). 



Taking the first of those equations, we have 



sin=e = 



P'+ Q= 



which 

 unity. 



2 (P- + Q) - R' ' 

 vill give a value of 9 between and 90°, provided the fraction be positive and less than 



Now the difference between the numerator and denominator 



= P'+ Q' - R\ 

 and A, B, and C being taken in order of magnitude, and A the greatest, Q (= A - C) is greater tlian 

 R {— B - C). Consequently P^ + Q' — R^ is positive, and the denominator of the above fraction 

 is positive and greater than the numerator, and sin 9 is possible. The value of r corresponding to 

 the value of 9 thus obtained, will be a minimum, and therefore PA will in this case be a maximtim 

 Hence it appears that PA is a maximtim value of the radius vector witli reference to the section of 

 the surface by a plane through the axis of .r-, while it is a minimum with reference to the section, 

 perpendicular to the former, made by the plane of yz. In this case then PA is neither a maximum 

 nor minimum value of the radius vector of the surface. 



Exactly the same conclusion may be drawn from the third of equations (/), in which case the 

 two sections to which PA is common, and one through the axis of x, and that made by the 

 plane of n-y. 



The annexed figure (2) represents the curve in each of the above cases referred to r and 0, 

 CPC' being in the first case, the axis of x, and in the 

 second the axis of z. PB and PB' represent the two 

 minima radii vectores in these sections. 



It remains to consider the second of equations (/), 

 which gives 



... P' + R" 



?,\n 9 = = s . 



2 {P' + R') - Q' 



Here, the denominator - the numei-ator = P^ + R' - (^. 



Now P=Q- R, (Art. 4) ; 



.-. P" + R' - Q' = 2R' - 2RQ 



= -2R{Q-R), 



which, since Q is greater than R, shews that the deno- 

 minator is less than the numerator. Consequently there 

 is no value of d between and 90', in this case, which 



dr 

 renders -73 = 0, and PA is here a minimum value of 



the radius vector, i.e. in the section made by a plane through the axis of y. PA is also a minimum 

 for the section made by the plane of arz. Consequently if figure (1) represent the plane of xz, each 

 of the four equal lines PA is an absolute minimum value of the radius vector of the surface, 



and ——J represents the absolute maximum value of the tangential force. The positions of these 



lines correspond to the following system of values of a ^ and y, 



/3 = 90", y = a = ± 45", 



