THE DESCENT OF GLACIERS. The following are the results of recent experiments* on the expansion of ice : Linear Expansion of Ice for an Interval of 100° of the Centigrade Thermometer. 0.00524 Schumacher. 0.00518 Moritz. Ice, therefore, has nearly twice the expansibility of lead; so that a sheet of ice would, under similar circumstances, have descended a plane similarly inclined, twice the distance that the sheet of lead referred to in the preceding article descended. Glaciers are, on an increased scale, sheets of ice placed upon the slopes of mountains, and subjected to atmospheric variations of temperature throughout their masses by variations in the quantity and the temperature of the water, which, flowing from the surface, everywhere percolates them. That they must from this cause descend into the valleys, is therefore certain. That portion of the Mer de Glace of Chamouni which extends from Montanvert to very near the origin of the Glacier de Léchaud has been accurately observed by Professor James Forbes. Its length is 22,600 feet, and its inclination varies from 4° 19′ 22′′ to 5° 5' 53". The Glacier du Geant, from the Tacul to the Col du Geant, Professor Forbes estimates (but not from his own observations, or with the same certainty) to be 24,700 feet in length, and to have a mean inclination of 8° 46′ 40". According to the observations of De Saussure, the mean daily range of Reaumur's thermometer in the month of July, at the Col du Geant, is 4°.2571, and at Chamouni 10092. The resistance opposed by the rugged channel of a glacier to its descent cannot but be different at different points, and in respect to different glaciers. The following passage from Professor Forbes's work contains the most authentic information I am able to find on this subject. Speaking of the Glacier of la Brenva he says:-"The ice removed, a layer of fine mud covered the rock, not composed, however, alone of the clayey line stone mud, but of sharp sand derived from the granitic moraines of the glacier, and brought down with it from the opposite side of the valley. Upon examining the face of the ice removed fro contact with the rock, we found it set all over with sharp angular fragments, from the size of grains of sand to that of a cherry, or larger, of the same species of rock, and which were so firmly "Vide Archir. f. Wissenschaftl. Kunde v. Russland, Bd. vii. s. 888. fixed in the ice as to demonstrate the impossibility of such a surface being forcibly urged forwards without sawing any comparatively soft body which might be below it. Accordingly, it was not difficult to discover in the limestone the very grooves and scratches which were in the act of being made at the time by the pressure of the ice and its contained fragments of stone." (Alps of the Savoy, pp. 203-4.) It is not difficult from this description to account for the fact that small glaciers are sometimes seen to lie on a slope of 30° (p. 35.). The most probable supposition would indeed fix the limiting angle of resistance between the rock and the under surface of the ice set all over, as it is described to be, with particles of sand and small fragments of stone, at about 30°; that being nearly the slope at which smooth surfaces of calcareous stone will rest on one another. If we take then 30° to be the limiting angle of resistance between the under surface of the Mer de Glace and the rock on which it rests, and if we assume the same mean daily variation of temperature (4-257 Reaumur, or 5·321 Centigrade) to obtain throughout the length of the Glacier du Geant, which De Saussure observed in July, at the Col du Geant; if, further, we take the linear expansion of ice at 100° Centigrade to be that (00524) which was determined by the experiments of Schumacher, and, lastly, if we assume the Glacier de Geant to descend as it would if its descent were unopposed by its confluence with the Glacier de Lechant; we shall obtain, by substitution in equation (2.) for the mean daily descent of the Glacier du Geant at the Tacul, the formula The actual descent of the glacier in the centre was 15 feet. If the Glacier de Léchaut descended, at a mean slope of 5°, singly in a sheet of uniform breadth to Montanvert without receiving the tributary glacier of the Taléfre, or uniting with the Glacier du Geant, its diurnal descent would be given by the same formula, and would be found to be '95487 feet. Reasoning similarly with reference to the Glacier du Geant; supposing it to have continued its course singly from the Col du Geant to Montanvert without confluence with the Glacier de Léchaut, its length being 40,420 feet, and its mean inclination 6° 53', its mean diurnal motion 7 at Montanvert would, by formula (2.) have been 2.3564* feet. The actual mean daily motion of the united glaciers, between the 1st and the 28th July, was at Montanvert,†t *On the 1st of July the centre of the actual motion of the Mer de Glace at Montanvert was 2.25 feet. Forbes'" Alps of Savoy," p. 140. The motion of the Glacier de Léchaut was therefore accelerated by their confluence, and that of the Glacier du Geant retarded. The former is dragged down by the latter. I have had the less hesitation in offering this solution of the mechanical problem of the motion of glaciers, as those hitherto proposed are confessedly imperfect. That of De Saussure, which attributes the descent of the glacier simply to its weight, is contradicted by the fact that isolated fragments of the glacier stand firmly on the slope on which the whole nevertheless descends. It being obvious that if the parts would remain at rest separately on the bed of the glacier, they would also remain at rest when united. That of Professor J. Forbes, which supposes a viscous or semi-fluid. structure of the glacier, is not consistent with the fact that no viscosity is to be traced in its parts when separated. They appear as solid fragments, and they cannot acquire in their union properties in this respect which individually they have not. Lastly, the theory of Charpentier, which attributes the descent of the glacier to the daily congelation of the water which percolates it, and the expansion of its mass consequent thereon, whilst it assigns a cause which, so far as it operates, cannot, as I have shown, but cause the glacier to descend, appears to assign one inadequate to the result; for the congelation of the water which percolates the glacier does not, according to the obser vations of Professor Forbes,* take place at all in summer more than a few inches from the surface. Nevertheless, it is in the summer that the daily motion of the glacier is the greatest. The following remarkable experiment of Mr. Hopkins of Cambridge, which is considered by him to be confirmatory of the sliding theory of De Saussure as opposed to De Charpentier's dilatation theory, receives a ready explanation on the principles which I have laid down in this note. It is indeed a necessary result of them. Mr. Hopkins placed a mass of rough ice, confined by a square frame or bottomless box, upon a roughly chiselled flag-stone, which he then inclined at a small angle; and found that a slow but uniform motion was produced, when even it was placed at an inconsiderable slope. This motion, which Mr. Hopkins attributed to the dissolution of the ice in contact with the stone, would, I apprehend, have taken place if the mass had been of lead instead of ice; "Travels in the Alps," p. 413. I have quoted the following account of it from Professor Forbes's book p. 419. and it would have been but about half as fast, because the linear expan sion of lead is only about half that of ice. NOTE G. THE BEST DIMENSIONS OF A BUTTRESS. Ir m, (Art. 299.) represent the modulus of stability of the portion AG <1 the wall, it may be shown, as before, that P{(hh) sin. a—(l—a—m,)cos. a} = (a,m) (h, — h2)a‚μ; ... P{(h, — h2)sin. a — (l — a,)cos. a} = 1(h, — h2)a,3μ―m, {P cos. a + (h, — h ̧)α‚μ} If m=m, the stability of the portion AG of the structure is the same with that of the whole AC; an arrangement by which the greatest strength is obtained with a given quantity of material (see Art. 388.). This supposition being made, and m eliminated between the aboɣe equation and equation (388.), that relation between the dimensions of the buttress and those of the wall which is consistent with the greatest economy of the material used will be determined. The following is that relation: It is necessary to the greatest economy of the material of the Gothic buttress (Art. 301.) that the stability of the portions Qa and Qb, upon their respective bases ac and be, should be same with that of the whole buttress on its base EC. If, in the preceding equation, h1-h, be substituted for h, and h-h, for h, the resulting equation, together with that deduced as explained in the conclusion of Art. 301., will determine this condition, and will establish those relations between the dimensions of the several portions of the buttress which are consistent with the greatest economy of the material, or which yield the greatest strength to the structure from the use of a given quantity of material. NOTE II. DIMENSIONS OF THE TEETH OF WHEELS. THE following rules are extracted from the work of M. Morin, entitled Aide Mémoire de Mécanique Pratique :-If we represent by a the width in parts of a foot of the tooth measured parallel to the axis of the wheel, and by its breadth or thickness measured parallel to the plane of rotation upon the pitch circle; then, the teeth being constantly greased, the relation of a and b should be expressed, when the velocity of the pitch circle does not exceed 5 feet per second, by a=4b; when it exceeds 5 feet per second, by a = 56: if the wheels are constantly exposed to wet, by a= - 6b. These relations being established, the width or thickness of the tooth will be determined by the formulæ contained in the columns of the following table: Assuming that when the teeth are carefully executed the space between the teeth should be th greater than their thickness, and th greater when the least labor is bestowed on them, the values of the pitch T will in these two cases be represented by b(2+) and b(2+), or by 2:0676 and 2:16. Substituting in these expressions the values of b given by the formula of the preceding table, then determining from the resulting values of c (see equation 233.) the corresponding values of the coefficient C (see equation 234.), the following table is obtained: |