156 Mr. Hopkins on the Mechanism of Glacial Motion. 



directions are also at right angles to each other. Conse- 

 quently the maximum and tninimum values ofy cannot differ 

 in intensity, and only therefore in sign. 



Fig. 3. 



^y 



^ 



12. From these results, 

 with respect to R andyj we 

 have the following conclu- 

 sions : — Let q be any point 

 in the mass in a state of 

 constraint; then if R ^ R 

 be the direction of maxi- 

 mum tension at (7, r*//' per- 

 pendicular to R (7 R, will 

 be that of minimum ten- 

 sion; and if the two lines 

 fqf be perpendicular to 

 each other, and make an- 

 gles of 45° with R q R and 

 rqr, they will be the direc- 

 tions in which the intensity 

 of y will be the greatest. It should also be recollected that 

 /■= in the two directions R t^ R and rqr. 



There is also a simple relation between the maximum value 

 (/) ofy^ and the maximum and minimum values (R) and (?') 

 of R. It is obtained immediately from equations (3.) and (5.), 

 which give 



(R) -(;•)■-= 2(7). 



13. If we take a rectangular element at q^ of which the sides 

 are respectively parallel to RR and ?r, it will be acted on by 

 no tangential forces, and therefore will be held in equilibrium 

 by the maximum and minimum normal tensions alone. Con- 

 sequently, whatever may be the forces acting on the mass, the 

 state of tension or pressure at any point {q) will be the same 

 as if the forces impressed on that point were those of two sy- 

 stems of forces, of which the intensities should be (R) and (r), 

 and acting in directions perpendicular to each other. This 

 proposition is important, as leaving no doubt as to the direc- 

 tion in which the greatest tendency is exerted by the normal 

 forces to fracture the mass, a point to be considered in the 

 sequel. 



That there may be no doubt as to the equivalence of forces 

 asserted in this })roposition, let us assume the forces impressed 

 on the mass at q to be (R) and (r), instead of Xj, Yj and^J, 

 and thence deduce the expressions for liie forces acting nor- 

 mally and tangentially on the line q& (fig. 2). Calling these 

 forces X and yj and considering (R) and (/•) as tensions^ we 

 have 



