150 Mr. Hopkins on the Mechanism of Glacial Motion. 



and therefore 



mn |3 . ^ . 

 ^rj^ = -f- sin 2 5. 

 MN 2a 



Consequently the extension is a maximum when 3 = 45°; also 

 it is a minimum when 9 = 135°, or the compression is a maxi- 

 mum for this latter value of 9. 



6. To pass generally from the extension and compression 

 to the corresponding values of the tension and pressure, it is 

 necessary to know the laws which connect them ; but here we 

 are not concerned with this more general question, our object 

 being merely to deduce the directions of maximum tension 

 and pressure from those of maximum extension and compres- 

 sion. Now, whatever may be the general law connecting ten- 

 sion and extension, there can be no doubt that when the one 

 is a maximum or minimum the other must be so likewise. 

 Hence the directions above determined will be those of max- 

 imum tension and 'pressure, so far as the tension and pressure 

 are superinduced by the motion of the glacier. Consequently 

 if M N and M N' be those directions for the point M, each will 

 be inclined at an angle of 45° to the direction of motion, and 

 they will be at right angles to each other. 



The preceding method has the advantage of pointing out 

 very simply and distinctly the manner in which the difference 

 of velocity in diffei'ent parts of the glacier produces tension 

 and pressure, and the results above given are arrived at with 

 great facility. But I shall now proceed to a more complete 

 investigation of the problem, still preserving the same limita- 

 tion as before in the hypothesis of the equality of the veloci- 

 ties of the upper and lower surfaces of the glacier. Our re- 

 sults will thus be rendered independent of the depth of the 

 glacier, which may, therefore, be treated as a simple lamina. 

 We shall thus only be concerned with space of two dimen- 

 sions. 



7. Internal Tensions and Pressures of a Glacier. — When a 

 plain solid lamina, having a certain degree of compressibility 

 and extensibility, is brought into a position of constraint by 

 forces acting in the plane of the lamina, the particles on one 

 side of a geometrical line will exert certain forces on the con- 

 tiguous particles on the opposite side of the line. If the lamina 

 were formed of fluid particles the resultant action at each point 

 of this line of separation would be normal to it; but when the 

 lamina is solid this will not be generally the case, and there- 

 fore the force at any point of the line may be resolved into two 

 forces, one being normal and the other tangential to the line 

 of separation ; all forces being supposed to act in the plane of 

 the lamina. Suppose the line of separation to be a straight 



