Mr HOPKINS, ON RESEARCHES IN PHYSICAL GEOLOGY. 15 



the tendency of these forces to separate the contiguous particles on 

 opposite sides of the elementary portion Sx, of the geometrical line 

 AB, estimated by their tendency to give an opposite motion to these 

 particles along any assigned line rPR, we must resolve the forces in the 

 direction of that line. Let BPQ^O; then MdU the sum of the resolved 

 parts of our forces in the directions PR and Pr be 



^.r./^sinv/,cos() + ^.r./sin (x//-/3) cos(/3-e) (A). 



If the value of this expression, considered as a function of the inde 

 pendent variables ^ and e be made a maximum, we shall manifestly 

 obtain from the corresponding value of ^ that angular direction of the 

 line AB along which the two sets of tensions we are considering have 

 the greatest tendency to form a fissure. 



Differentiating the expression with regard to 6, we have 

 Sx . Fsin ^j.sm9- &x ./sin («/, - /3) sin (/3 - tJ) = o. 



The left-hand side of this equation is the expression for the sum 

 of the resolved parts of the forces ^x.F^m^ and ^xf^mU-Q) per 

 pendicular to the line PR. Consequently the equation expresses the 

 condition that PR must coincide with the direction of the resultant of 

 the above forces. 



Again, differentiating with respect to x/,, we obtain 



Feosf. COS0 +/COS (x/, - /3) cos ((3-6) = 0. 

 From the above equations we must determine x/, and 0. If we 

 ~p = fi, we obtain from thence 



1 +M (cos^- sin/3cotx//) (cos/3- sin /3 . cot0) = o, 



1 + M (cos/3 + sin/3 . tan>/.) (cos^ + sin/3 tan 9) = o, 



or, putting cos/3 = c, sin/3 = *, cot9 = x. cot^l. = x, 



1 +fx{c- sz){c- sx) = 0, 



'-(«+D("i)-<'- 



put 



