RIGIDITY OF BOCKS AND HYSTERESIS FUNCTION. 23 



Lei us now investigate some general properties <>f this hysteresis 

 function in the following pages ; — To begin with, it is necessary to 

 remark that the strain corresponding to any stress is here supposed 

 to consist of two terms, of which the principal term is simply pro- 

 portional to the actual stress while the other is given by the hysteresis 

 function, r t • thus 



r=r + 3? . (9). 



Let the specimen be freed from any external couple after it has 

 been once acted upon by a couple whose amount is given by p, so 

 that in the equation (4) n=p i and then let it be again acted upon by 

 a couple. The yielding apparently due to unit couple, no matter 

 whether it be positive or negative, is given by 



iJ*L) = k log- r - VP + r + ] L__ = - ,„ say (10). 



which is always negative. Now, if the newly applied couple be 

 negative, the principal term due to this negative couple is necessarily 

 negative ; let it be —~\ so that the increase of twist apparently 

 due to this negative unit couple is 



r ft =-{r> + r 3 ] (11). 



On the other hand, if it be positive, the principal term is also positive 

 and, by Hooke's law, equal to +n ; so that the increase of twist due 

 to this positive couple is 



V^-r, (12). 



Thus, the absolute value of r„ being greater than that of t p , we 

 have following interesting result : 



Proposition I. If after withdrawing die whole couple applied to a 

 piece of rock, we begin to reapply it, the specimen mast apparently be more 

 rigid in one direction than in the other. 



In the equation (5) put_p = ?i, and we have 



a = klon- r&P+'+t+l } H'+l} . (l3) 



9 r[p+r+t+i]r[p+t+i} (id) - 



