to the right with the velocity a(1-e) represented by the equations (1+e)v=F{x—a(1—e)t} + a(1+e) [1-ƒ{x-a(1-e)t}, as=(1+e)v; while to the left will be propagated, with the velocity a(1+e), a disturbance represented by (1—e)v=†F{x+a(1+e)t} _a(1−c) [1−ƒ{x+a(1+e)t}], -as= = (1-e)v. 2 It only remains, therefore, to determine under what circumstances (12), and under what circumstances (18) is to be taken as the equation of motion. Comparing (11) with (1), we get Now it is obviously impossible that in any particular case of motion Ρ should have two values. We have therefore to determine in each particular case of motion which of the above values is to be taken. At a Suppose that we have at a given epoch, in two different tubes, exactly the same kind of disturbance, with this difference, viz. that the velocity at each point of the one is in the opposite direction to that at the corresponding point of the other. point in each for which the values of v and s are identical except as regards the sign of the former, it is clear that the pressure must have the same absolute value; but it is equally clear that the expression for the pressure must differ in the two cases. If in the one case, the particle-motion being to the right, and x being measured positively in the same direction, the pressure is represented by p=+2Daev + Da2(1+-s), then in the other case, the particle-motion being to the left, a being measured as before, the pressure must be represented by and vice versa. p=-2Daev + Da2 (1+s), The occurrence of the double sign in the value of p and in the (11) is thus at once accounted for. It still remains to be determined, however, whether, when the particle-motion at a given point is to the right, the coefficient of v in the expression for the pressure should have a positive or negative sign affixed to it. Suppose that, the air being of uniform density and at rest, a disturbance is suddenly impressed upon a limited portion of it of this kind, viz. a velocity which beginning at zero gradually increases till it attains the value v1, and thence diminishes by the same gradations till it finally vanishes, the density throughout being unaffected. According to the formula (19), we shall have under these circumstances, p=Da2+2Daev, i. e. the pressure will be either increased or decreased by the impressed velocity. The sign to be attributed to the coefficient of v in the last expression, equally with the numerical value of the constant e, is a proper subject for experiment; nevertheless I think we may conclude with perfect certainty that the lower sign is to be taken (in other words, that the effect of the impressed velocity is to diminish the pressure in the portion of the air affected by it), on the following grounds. We have before us the following alternative. For the sake of perspicuity, assuming that the particle-motion thus supposed to be impressed tends to the right, we must either have the pressure gradually increasing as we move from the left of the disturbance till we reach its middle point, and thence gradually diminishing until it again assumes at the right-hand extremity the value of equilibrium, or else the pressure will diminish from the left-hand extremity up to the middle point, and will thence increase till it ultimately regains the value of equilibrium. But in either case we shall have in the first half of the disturbance (beginning from the left) the particles in each element flying from each other, the tendency, by reason of the impressed velocity, being to expansion; while in the second half the particles in each element are moving towards each other, the tendency in this latter case being towards compression. It appears, therefore, that we have to choose between two things, viz. on the one hand a diminished pressure where there is throughout a tendency to expansion, and an increased pressure where there is throughout a tendency to condensation; or, on the other hand, an increased pressure where there is throughout a tendency to expansion, and a diminished pressure where there is throughout a tendency to condensation. That this latter alternative should be true appears incredible. We may with safety conclude, therefore, that when the motion is to the right, x being measured positively in that direction, the lower sign is to be taken in (19), and vice versâ. Applying this conclusion to the results previously obtained, it follows that, when the motion is represented by (12), the particle-motion is to the left, and v is negative; so that any disturbance propagated to the right of the original disturbance will be a rarefaction, and its velocity of propagation will be a(1+e); while any disturbance propagated to the left must be a condensation, whose velocity of propagation will be a(1-e). On the other hand, when the motion is represented by (18) the particle-motion takes place to the right, i. e. v is positive; so that any disturbance propagated to the right of the original disturbance will in this case be a condensation, whose velocity of propagation is a(1-e); while any disturbance propagated to the left must be a rarefaction, and its velocity of propagation will be a(1+e). It results on the whole, therefore, that waves of condensation are propagated with the velocity a(1-e), which is less than what has hitherto been regarded as the calculated velocity apart from temperature; while waves of rarefaction are propagated with a velocity a(1+e), which is just as much greater than such calculated velocity. If it be asked whether is e so small that the difference between these two velocities is imperceptible to the human ear under all circumstances, or are two perceptibly distinct waves in fact propagated? I answer that e is not so small as that the difference between a(1+e) and a(1-e) cannot be distinctly appreciated. Two waves will in fact be propagated, one of which (the slower) the human ear is so constructed as to suppress. The proof of this I reserve for a future communication. 6 New Square, Lincoln's Inn, February 16, 1869. XXVI. On the Physical Cause of the Motion of Glaciers. I "All the parts," he remarks, " of a glacier do not descend with a common motion; it moves faster at its surface than deeper down, and at the centre of its surface than at its edges. It does not only come down bodily, but with different motions of its different parts; so that if a transverse section were made through it, the ice would be found to be moving differently at every point of that section... . . . . There is a constant displacement of the particles of the ice over one another and alongside one another, to which is opposed that force of resistance which is known in mechanics as shearing-force." He determines by calculation the amount of shearing-force which must not be exceeded if the displacement of the particles is to be effected by the weight of the ice alone. In the case of the Mer de Glace at the Tacul, the shearing-force of the ice must not exceed 1.3193 lb. per unit surface of one square inch, if that glacier descends merely by its weight, at the rate observed by Professor Tyndall. From experiments which he has made, he finds that the actual shearing-force of ice per unit surface is about 75 lbs. Consequently he concludes it is impossible that the motion of the glacier can be due to its weight alone; there must be some other force in addition to the weight impelling the ice forward. And he calculates that the amount of work performed by this unknown force is thirty-four times the amount performed by the weight of the glacier. This is a most important conclusion. It is quite decisive against the generally received opinions regarding the descent of glaciers by their own weight. But although it is thus demonstrated that glaciers cannot descend by means of their weight alone in the manner generally supposed, still, I venture to think that, notwithstanding the demonstration, gravitation after all may be the only force moving the ice. * Communicated by the Author. + Proceedings of the Royal Society, vol. xvii. p. 202. [See p. 229 of our present Number, ED. Phil. Mag.] The correctness of the above conclusion, that the weight of the ice is not a sufficient cause, depends upon the truth of a certain element taken for granted in the reasoning, viz. that the shearing-force of the molecules of the ice remains constant. If this force remains constant, then Canon Moseley's conclusion is undoubtedly correct, but not otherwise; for if a molecule should lose its shearing-force, though it were but for a moment, if no obstacle stood in front of the molecule, it would descend in virtue of its weight. The fact that the shearing-force of a mass of ice is found to be constant does not prove that the same is the case in regard to the individual molecules. If we take a mass of molecules in the aggregate, the shearing-force of the mass taken thus collectively may remain absolutely constant, while at the same time each individual molecule may be suffering repeated momentary losses of shearing-force. This is so obvious as to require no further elucidation. The whole matter, therefore, resolves itself into this one question, as to whether or not the shearing-force of a crystalline molecule of ice remains constant. In the case of ordinary solid bodies we have no reason to conclude that the shearing-force of the molecules ever disappears, but in regard to ice it is very different. If we analyze the process by which heat is conducted through ice, we shall find that we have reason to believe that while a molecule of ice is in the act of transmitting the energy received (say from a fire), it loses for the moment its shearing-force if the temperature of the ice be not under 32° F. If we apply heat to the end of a bar of iron, the molecules at the surface of the end have their temperatures raised. Molecule A at the surface, whose temperature has been raised, instantly commences to transfer to B a portion of the energy received. The tendency of this process is to lower the temperature of A and raise the temperature of B. B then, with its temperature raised, begins to transfer the energy to C. The result here is the same; B tends to fall in temperature, and C to rise. This process goes on from molecule to molecule until the opposite end of the bar is reached. Here in this case the energy or heat applied to the end of the bar is transmitted from molecule to molecule under the form of heat or temperature. The energy applied to the bar does not change its character; it passes right along from molecule to molecule under the form of heat or temperature. But the nature of the process must be wholly different if the transference takes place through a bar of ice at the temperature of 32°. Suppose we apply the heat of the fire to the end of the bar of ice at 32°, the molecules of the ice cannot possibly have their temperatures raised in the least degree. How, then, can molecule A take on, under the form |
