equation. Hence this equation cannot in general be satisfied except for a very small value of . small value of . If the surface were horizontal, we should have k = 0, cos & = O, and therefore 2qq. This relation appears to be nearly satisfied between mercury and glass, since the experiment of Casbois shows, that by merely boiling the mercury, its upper surface in a capillary glass tube may become horizontal, or even concave. The reasoning by which was shown to be in general a very small angle, does not apply to mercury, which is incapable, like water and oils, of adhering to a solid. The smallness of the angle of actual contact, seems to be a condition always satisfied, whenever a fluid moistens a solid*.

It is not possible to determine theoretically the form of the curve A NQ near the solid, because the laws of the molecular forces are unknown. Considering, however, the small extent of their sphere of activity, we may say that for a small distance the curve will be the same, whether the fluid rise against a plane surface or in a capillary tube. Let N be a point of the fluid surface situated just beyond the sphere of sensible action of the solid, let NR be drawn vertically, and the angle Q N R = w. It is proved, on the supposition of incompressibility, (see Poisson's New Theory of Capillary Action, art. 18,) that the quantity of fluid raised in a given tube varies as cos w. Now, as it is shown above that the angle of actual contact is constant, and very small, it follows that the angle w, and consequently the heights of 'ascent, may be different for different relative positions of A and N. If the fluid ascend in a tube not previously wetted, the points A and N will be nearest each other, w will have its greatest value, and the height of ascent be least. As A and N are more removed from each other by wetting more of the solid surface, the rise of the fluid may be expected to be greater, and to attain its maximum value when the moistening is carried to such a degree, that the solid can retain no more fluid attached to it. In this case the influence of the solid on the form of

* The views which this communication is intended to explain are stated, in part, at the close of my Report on Capillary Attraction (contained in the Fourth Report of the British Association), to which I take this opportunity of referring for the sake of pointing out an error in the remarks (p. 273.) on an equation of Laplace's theory, equivalent to that obtained above, excepting that gravity is not taken into account. It is not true, as there stated, that Laplace neglects the superficial variation of pressure. The strictures of Dr. Young upon it, adduced at p. 266 of the Report, will appear to be inapplicable from the reasoning above, which it is hoped will serve to place the inferences to be drawn from this equation, and its importance in the capillary theory, in a true point of view.

the fluid surface will be the least possible, and the column may be considered to be supported by the action of the fluid on itself. The fluid will thus hold the place of the solid in the equation previously obtained; we shall have q' = q, and the equation will become

q sing cos & + k.

Hence, on account of the smallness of g and k compared to q, the angle of contact with the aqueous tube will be small as well as of that with the solid. Consequently the intermediate angle will be small, and the heights of ascent be the same whatever be the solid, if the fluid thoroughly wets it. In the experiments of M. Link, (see Poggendorff's Annalen 1833, p. 404,) the condition here supposed was fulfilled by means of an apparatus for dipping the solid repeatedly in the fluid. The height of ascent is found to be independent of the nature of the solid, or nearly so. In subsequent experiments, (Poggendorff's Annalen, 1834, p. 593,) the plates between which the fluids ascended were previously dipped in strong caustic alkali and concentrated sulphuric acid, to get rid of a film of grease attaching to them, which they contract in the act of polishing. Water, in these new experiments, stood very nearly at the same height between glass, copper, and zinc plates. Other fluids did not follow this law, and in the instance of sulphuric acid, the deviation appeared to depend on a chemical action between the solid and fluid. Such a circumstance would be likely to affect the form of the curve ANQ and angle w, and consequently, according to the theory, the height of ascent.

The general inference from the foregoing reasoning is, that the heights of ascent do not merely depend on the molecular attractions, but, while these remain the same, may be affected by any circumstance that alters the form of the fluid surface near the solid, and particularly by the manner and degree of moistening the solid by the fluid. In this way the differences of the heights, as determined by different experimenters, may be accounted for.

The maximum height of ascent in a given tube varies, according to theory, for different fluids, as a certain quantity , in which g is the specific gravity of the fluid, and H* de

H

H

This is the quantity called H by Poisson, and by Laplace in his Supplementary Treatise. The quantity so denominated in Laplace's first Treatise is equal to This letter is inadvertently used sometimes in one of these senses, and sometimes in the other, in the Report on Capillary Attraction.

pends on the law, intensity, and sphere of activity of its molecular forces. Admitting the superficial variation of density to be negligible, and the fluids to be incompressible, H will be a measure of their cohesiveness, or the inverse of it a measure of their fluidity, as it is proportional to the effect of their molecular attractions acting under the same circumstances. (Laplace's Treatise, art. 12. and Supplement, p. 18.) On this account it formerly appeared to me simplest to suppose this quantity to vary as g; and the first experiments of M. Link favoured this idea by assigning to different fluids the same height of ascent. Those subsequently made, which are the more accurate, do not give the same result, but sufficiently prove that the heights are not as the specific gravities, and consequently that H does not vary as g, as is usually supposed. Probably nothing can be determined respecting it à priori. The last-mentioned experiments gave the following results when the fluids ascended between glass plates thoroughly moistened:-height x specific gravity = 5.3, for sulphuric æther; 67, for alcohol; 10.7, for liquid caustic alkali; 10.9, for liquid ascetic [carbonated?] alkali; 12.5, for water; 156, for muriatic acid; 16.8, for nitric acid; 20'3, for sulphuric acid. According to what is said above, these numerical quantities are in the order of cohesiveness, sulphuric æther being the least cohesive body, or possessing the greatest degree of fluidity.

The phænomenon of endosmose may be appealed to as indicating a great attractive energy in partially fluid substances. When water is on one side of the porous membrane, and an imperfect fluid, as treacle, or solution of gum, on the other, the latter is found to draw the water powerfully through the pores. The force exerted at any time appears by experiment to be in this case proportional to the difference of the densities of the fluids on the opposite sides of the membrane.

The subjects treated of in this paper are of such a nature as scarcely to admit of any very definite discussion. Since, however, the degree in which capillary phænomena may be affected by a variation of density at the surfaces of fluids is at present quite unknown, it seemed desirable at least to ascertain whether an approximation to the truth is obtained when that variation is neglected; and, perhaps, the preceding reasons, connected with the nature of fluidity, may make it probable that such is the case.

Papworth St. Everard, Dec. 10, 1835.

XVII. Letter from Peter Barlow, Esq., F.R.S., to the Rev.
D. Lardner, LL.D., F.R.S., on the Theory of Gradients in
Railways.

S

DEAR SIR,

As you have addressed a letter to me in the last London and Edinburgh Philosophical Magazine, I feel myself bound by courtesy to reply to it through the same Journal, not, however, with a view of entering into any controversy on the question. You have in your letter very clearly stated your mode of solution, I will endeavour to explain also the grounds of my objection; the readers of the Lond. and Edinb. Philosophical Magazine will then be able to form their own judge

ment.

First, t being the fraction expressing the ratio of the friction to the load, and the angle of the plane's inclination, you take t to denote the resistance on the horizontal plane, t + sin to denote the same on the ascending plane, and t- sin for the same on the descending plane. Then, assuming what may not, perhaps, be quite true, (but to which I do not here make any objection,) i. e. that in locomotive engines the power generated and expended is the same in the same time, you arrive at the conclusion, that in cases of uniform velocity the resistance into the velocity is constant: taking, therefore, V to denote the velocity on the horizontal plane, and ʊ the velocity on the descending plane (which you also assume to be uniform), you arrive at the equation

(t sine) v tV.

t

sin

And by this formula your tables of velocities are computed for the Great Western and Basing lines; but where the formula gives more than 40 miles an hour the results of the computation are not stated.

Thus, for example, assuming, as you do, 25 miles to be the velocity on a horizontal plane, and t =

1

250'

the formula be

(004-sin ε)

I have gone over all the numbers in your four tables, and, except very trifling numerical errors, I find your results consistent with this formula as far as you have given the computed velocities; but those which you have not stated (by

supposing the break to be employed) are so extraordinary that I think you can scarcely consider them as correct.

For example, taking a very common gradient of 16 feet to

004

= 103.09 miles

per

330

hour.

Such a gradient is very common in practice, and in practice in such a case the break is not applied. The computed and practical result ought, therefore, to agree, supposing the solution correct; but such a velocity as one of 103 miles per hour has never yet, I believe, been obtained.

Again, with a slope of 1 in 250, by no means an uncommon gradient, your formula gives the velocity of descent infinite: now such gradients are descended without the break, but, of course, not with an infinite velocity. For slopes greater than the last, of which also there are many, your velocity passes through infinity, and becomes negative, and the time of descent negative also, or less than no time.

I have certainly stated in my Second Report to the Directors of the London and Birmingham Railway Company, that a solution which leads to such extraordinary results must be "erroneous both in theory and practice." And my opinion is not altered. The error, I conceive, arises from combining the two dissimilar forces t and sin e, and then treating the question of descent as one belonging to the case of uniform m tions; whereas (according to my view of the subject) it pre perly belongs to the class of accelerated motions.

Your solution, however, and my objection to it are thus placed before the readers of the London and Edinburgh Philosophical Magazine, and I leave the question in their hands. I remain, dear Sir, yours truly,

Woolwich, Jan. 2, 1836.

PETER BARLOW.

After closing my letter, I have thought it might be satisfactory to some of the readers of the Phil. Mag. to see the view I take of this question, which is as follows.

Suppose a body free from friction to arrive at, or to be propelled from, the upper end of an inclined plane with a velocity, and let the angle of the plane, then the velocity acquired by that body in the time t will be v + 2 g. sin e.t; and the space descended will be tv + g sin e. t; (g denoting the space fallen through by gravity in one second, or g = 16 feet.)

Suppose now a body subject to friction to reach the same plane with the same velocity, but that this body contains