Mr. Hopkins on the Mechanism of Glacial Motion. 149 



with the difference of velocity of M and N, when our object 

 is to calculate the extensions and compressions due to the 

 motion of the mass. For the greater distinctness, let us con- 

 ceive the mass to be at first in a state in which it is neither 

 extended nor compressed. In a given time, considering M at 

 rest, let N ti be the space due to the difference of motion of 

 M and N, or to the relative velocity of N. The physical line 

 whose unextended length was M N will now be extended into 

 M n ; and in like manner, if N' be to the left of L (M L being 

 perpendicular to p' q' and r's'), M N' will be cojnpressed into 

 M n'f where N' n' = N w. This shows distinctly the manner 

 in which the motion in question must necessarily produce ex- 

 tension and compression in the mass in different directions. 



If the space Nw due to the relative motion of N become 

 sufficiently great, the physical line M N will be so extended 

 as to break, and an open fissure will be the consequence. 

 Our first object, however, will be to find those directions in 

 which the extension and compression have their maximum 

 values, before the extension has become so great as to cause 

 the fracture of the mass. And here it is important to remark, 

 that so long as we keep within this limit, we have no concern 

 whatever with the cohesive force of the mass, or the law ac- 

 cording to which it may vary in passing from one point to an- 

 other. That consideration will only enter when we proceed 

 to the ulterior part of our investigation, where the object will 

 be to determine the directions in which the mass will be frac- 

 tured, when the internal tensions shall become greater than 

 those which its cohesive force is able to resist. 



5. The extension of the physical line M N will be correctly 



represented by the ratio ^ir^i the point m being so taken that 



M m shall = M N. This ratio will manifestly be different for 

 different lines, such as M N; our first object is to find the 

 position of M N for which the ratio will be a maximum, subject 

 to the condition that N?i shall be the same for all positions of 

 N. The problem, as thus presented to us, is not mechanical 

 but geometrical, and a very simple one. To render it still 

 more simple in the calculation, I shall here suppose N 7Z a 

 small quantity of the first order, and shall omit small quanti- 

 ties of the second and higher orders*. Let M L = «, N w 

 = |3, and L M N = 5 ; then shall we have, to the requisite 

 degree of approximation. 



m « = /3 sin 9, M N = 



a 



cosd 



* A more complete investigation, and independent of this restriction, will 

 be given afterwards. 



