IN so far as Professor Lliboutry is trying to make the theory of glacier flow more realistic one can only wish him well and hope that h e is on the right track. L eaving aside for the moment the question of the crevasses, my objections were entirely directed against his calculation made with a specific model, namely a perfectly plastic glacier, a nd here the questions are not ones of opinion, but of the truth or falsity of mathematical deductions. As a first approximation Lliboutry's formulae

more »

... r the glacier profile and the stresses are, as I submitted before , correct but not new , and on this we seem to be in agreement. The difficulties begin when we come to the second approximation. I said that the m ethod Lliboutry used appeared to me to be completely fallacious. I did not say why this was so because it seemed sufficient to point out that the resulting "solution" did not satisfy one of the boundary conditions: that the upper surface of the glacier should be free from shear stress. Llibou try now says that the figure (Fig. 5 of the original paper) which m akes this obvious should not have been drawn and insists that the second approximation for the profile remains unchallenged. It is therefore now necessary for me to say, as briefly as I can, why the method used is fallacious. It is a question of the correct way of m anipula ting approximate equations. Since the mistake is present whether the slope of the bed ~ is zero or n o t, I shall take the case ~=o for simplicity. The equation after ( 12) on p. 255 of the original paper is T = pg (e -z) tan rJ.. cos (3, which for (3 = 0 reduces to T = pg (e -z) tan rJ.. W e seem to be agreed that this is only a first approximation, in the sense that it approaches the true equation for T as rJ. approaches zero (it certainly cannot be true for large rJ.). The true equation for T must therefore be T = pge {(I -~) rJ.+ O(rJ.Z)}, ( I ) where O(rJ. l ) means terms of order rJ.l and higher, which will, in the absence of proof to the contrary, be functions of x and z. If Lliboutry thinks the t erms he omits are of order rJ.3 or higher rather than rJ. l , it is up to him to say so explicitly and to prove it. On the bed, z = 0 and T = To; whence the terms O(rJ. l ) being functions of x. Then, dividing (I) b y (2), we find as the full version of Lliboutry' s equation (13) . The upper surface (z =e) must be free from shear traction, but since the surface is not p erpendicular t6 the z axis, which is vertical, Twill not in general be zero at z=e. Thus we may notice that the terms ToO(rJ.) in equation (3) are essential for the satisfying of the upper boundary condition. It then follows that Lliboutry's equation (14) omits terms of the form To O(rJ.) . In order to substitute in the second equation of equilibrium, which is (with ~=o) ()u z + OT __ -pg, oz ox