Mr. HOPKINS, ON THE INTERNAL PRESSURE OF ROCK MASSES. 463 



111 each case I'A is a minimum value of the radius vector with reference to the hyperbola 

 of which it is the semi-axis major, and consequently PA is the position of the normal to the plane s, 

 when the tangential force upon it is, in the same relative sense, a maximum. It still remains 

 to be determined whether PA is also a mitiimiim value of the radius vector in a section of the 

 surface made by a jjlane tlirough PA and the co-ordinate axis perpendicular to the plane of 

 the paper. For this purpose, let this last-mentioned axis be first taken as that of ,r, and let 



w = r cos 0, y = f sin Q sin d), z = r sin Q . cos (p, 



r 9 and (p being the ordinary polar co-ordinates. Substituting these values in the equation (4.) 

 to tile surface, and putting (p = id" , we obtain 



P'+ Q' . , ^ (P- + Q' if- 



sin- 9 



— )Mn'e (5), 



the polar equation to the section through the axis of x and the axis of either hyperbola in the 

 plane of yz. 



Similarly, putting 



y = r cos 9, z = rsm9 sin (p, ,v = r sin 9 . cos (p, 

 we obtain 



-■^^.i»--(^-!)-» <«). 



the polar equation to the section of the surface by a plane through the axis of y and the axis of 

 either hyperbola in the plane of <vz. 



Again, putting 



sr = ?• cos 9, y = rsin 9 sin (p, .r = »• sin cos (p, 

 we iiave 



^■^n-'-«-(-T^*-T)-« «■ 



the e(iuation to the section through the axis of z^ and the axis of either hyperbola in the plane of .vy. 



cIt 

 Differentiating (.5), ((>), and (7), and putting -- = 0, we obtain in the several cases, 



du 



\(P' + Q-) - {2 (P- + Or) - R'\ sm'9} sin 9 cos 9 = 0, j 



{(i^-H R')- {siP"- + R') - Q'] sin' 9} sine cos 0=0, [ (/)■ 



{{Q' + R') - l2(Q- + R') - P} s\n-9\ sin cos 0= 0, I 



Each of these equations may be satisfied by 



sin = 0, cos 9 = 0. 



The first corresponds to r = es , the axis from which 9 is measured being an asymptote to the 

 curve. The second gives 9 = 90", and therefore r = AP, which is consequently either a maxiimini 

 or a minimum value of r with respect to the curve in which r and 9 are the variai)le co- 

 ordinates. Now since r = oo when 9 = 0, or 180", and r = AP wiien 9= !K)", it is manifest that 



dr 

 AP must be a minimum value of r, provided — is not rendered zero by any value of 9 



iiO 



dr 

 between and ^O"; but if, on the contrary, -— - become zero for some value of 9 l)ctween those 



d9 



limits, the corresponding value of r must bo a minitintrti, in which c.ise PA will be a maximum 

 Vol.. VIII. I'aht IV. .( O 



