414 R. M. Beeley — Viscous Flow of Glacier-Ice. 



But by drawing CD, G^D^ perpendicular to C A, C^ 4^ we have 

 by similar triangles 



^ _ ^ fa\ 



BC — BD ' • ' ' ^^^ 



75 p 



therefore ^^^ is proportional to B C, and B D must be a constant, 



wliich is a characteristic property of the parabola. Hence the 

 curve Ci GE-^ is a parabola whose axis is Y Y. 



So we find that when a viscous substance is flowing under the 

 action of gravity, or is propelled by a pressure uniformly distributed 

 over a section normal to the direction of flow, the shearing stress is 

 always greatest at the boundary. This is so even when the moving 

 mass passes over a hollow with gently sloping sides. In Fig. 2 we 

 may suppose that we are dealing with the flow of a very broad 

 glacier moving down an incline. Every particle of it is being 

 urged along in the direction of the inclination of the upper surface 

 with a force depending upon the mean slope of the surface area 

 immediately above it. The slope of the surface upon which it rests 

 is only of secondary importance. The motive power wholly results 

 from the inclination of the upper surface. If it were due to the slope 

 of the lower surface a mass of ice in a basin-shaped hollow would 

 form a vortex ring, and we should have a case of perpetual motion. 

 Indeed, we must regard a glacier as moving just as a river does 

 when the slope is very small (insufficient to make sinuous motion 

 take the place of direct motion), but owing to its greater viscosity as 

 moving at a much slower speed. 



At each cross-section the volume passing must be the same, pro- 

 vided no ice is melted at the surfaces. Where the ice is thickest the 

 slope of the upper surface is least, and where thinnest the slope is 

 greatest. That this is so can be seen from the simple equations 

 applicable where inertia does not influence the result. 



Writing B as the total force on the liquid we have. Fig. 2, 



B = Br,h .... (7) 



At the bounding plane XX, and making F the pressure on unit 

 area, we have from equations (1) and (4) 



Y 



jl — fj^ah -^ . . . (8) 



.-. V, = ^ ... (9) 



Making F the maximum velocity, as the curve of distortion is 

 a parabola, we have V — ^ V^, and from (9) and (7) 



7= -o^ . . . (10) 



_ ^''o . . . (11) 



'2i a jji 



