﻿232 Royal Society: — 



friction. A double integration of each of the functions, thus-repre- 

 senting the internal work in respect to a given elementary prism, 

 determines the whole internal work of the trapezoid, in terms of the 

 space traversed by the middle of the surface in one day, the spaces 

 traversed by the upper and lower edges of the side, and a symbol 

 representing the unit of shear. Well-known theorems serve to de- 

 termine the work of the shear and the friction of the bottom and 

 side in terms of the same quantities. All the terms of the equation 

 above referred to are thus arrived at in terms of known quantities, 

 except the unit of shear, which the equation thus determines. The 

 comparison of this unit of shear (which is the greatest possible, in 

 order that the glacier may descend by its weight alone) with the 

 actual unit of shear of glacier ice {determined by experiment), shows 

 that a glacier cannot descend by its weight only ; its shearing force 

 is too great. The true unit of shear being then substituted for its 

 symbol in the equation of condition, the work of the force, which 

 must come in aid of its weight to effect the descent of the glacier, is 

 ascertained. 



The imaginary case to which these computations apply, differs from 

 that of an actual glacier in the following respects. The actual 

 glacier is not straight, or of a uniform section and slope, and its 

 channel is not of uniform roughness. In all these respects the re- 

 sistance to the descent of the actual glacier is greater than to the 

 supposed one. But this being the case, the resistance to shearing 

 must be less, in order that the same force, viz. the weight, may be 

 just sufficient to bring down the glacier in the one case, as it does in 

 the other. The ice in the natural channel must shear more easily 

 than that in the artificial channel, if both descend by their weight 

 only ; so that if we determine the unit of shear necessary to the 

 descent of the glacier in the artificial channel, we know that the 

 unit of shear necessary to its descent by its weight only in the na- 

 tural channel must be less than that. 



A second possible difference between the case supposed and the 

 actual case lies in this, that the velocities of the surface-ice at differ- 

 ent distances from the edge, and at different heights from the bottom, 

 are assumed to be proportional to those distances and heights ; so 

 that the mass of ice at any time passing through a transverse section 

 may be bounded by plane surfaces, and have a trapezoidal form. 

 This may not strictly be the case. All the measurements, however, 

 show that if the surfaces be not plane, they are convex downwards. 

 In so far therefore as the quantity of ice passing through a given 

 section in a day is different from what it is supposed to be, it is 

 greater than it. A greater resistance (other than shearing) is thus 

 opposed to each day's descent, and also a greater weight of ice 

 favours it ; but the disproportion is so great between the work of 

 the additional resistance to the descent, and that of the additional 

 weight of ice in favour of it, that it is certain that any such con- 

 vexity of the trapezoidal surface would necessitate a further reduction 

 of the unit of shear, to make the weight of the actual glacier suffi- 

 cient to cause it to descend. 



