Mr HOPKINS, ON RESEARCHES IN PHYSICAL GEOLOGY. 17 



6. If the two systems of tensions be equal and perpendicular to 

 each other, equation (1) becomes 



siuTT . cot"\// — 2 (] + costt) cotx// — sin-n- = 0, 



and is satisfied independently of particular values of ^. In this case, 

 therefore, there is no greater tendency to form a fissure in one direc- 

 tion than another. If F be greater than^ the equation becomes 



sinTT. cot-x//— 2 ( Ij cot\|/-sin7r = 0, 



of which the two roots are and oc, which shews that the greatest 

 tendency is to form a fissure in a direction perpendicular to that of F". 



7. The above investigation easily admits of generalization for any 

 number of systems of parallel tensions superimposed upon each other. 

 Let F denote, as before, the intensity of the tension in the direction 

 from which 6 and ^p are measured; f„ f,, &c. the tensions in direc- 

 tions making respectively angles /3,, /3,, &c. with the direction of the 

 tension F. Then shall we have 



Sx. {Fsm^}^cose+f,sm{^}^-fi,)cos(e-|3,)+f,sm{f-(i,)eos{9-(3,)+&ic.} = max.■, 

 and proceeding exactly in the same manner as in the previous inves- 

 tigation, and adopting an analogous notation, we shall manifestly obtain 

 the following equations : 



1 + Ml (C, - «,S) (C, -SiX)+ Ms (Cj - S-iZ) (Co - S;X) + &c. = 0, 



and putting, for the same reason as before, x= , we obtain 



l+fi^{c^-SiZ)— — ^'-l-&c. = 0; 



.-. 2 + /.,{c,*, + (C|'-*,')s:-c,.s,s'J + &c. = 0; 



or — {m,c,«, +|U2C2«2 + &c.}«'' 



+ {1 +(i,(Ci'-*,') + ixiici' - St') + &ic.\ z + fiiCiSi + fiiC^Si + SiC. = 0; 

 Vol. VI. Paet I. C 



