unextended dimensions of each to have been the same. Again, suppose a third lamina superimposed in the same manner, and then a fourth, and so on, till a mass of any assigned thickness shall have been thus composed. It will then follow, from what has been shown, that the tension at any point c of the mass in this state must lie in the plane of the section, and in the direction to the tangent of the curve-line acb, formed by the intersection of the vertical plane of the section with the lamina in which the point c may be situated.

The only difference between this hypothetical mass and any proposed actual mass of the same form and dimensions, will consist in this—that in the former there is no cohesion whatever between the successive lamina of which we have supposed it to be formed. If, however, our laminæ should be superposed on each other in their unextended state, and made to cohere firmly together, (in which case the mass would differ in no wise from any actual mass,) and then elevated to the position represented in the diagram, it is easily seen that the position of each point of the mass would be exactly the same as in the hypothetical case above stated. Consequently, the extension of any portion of the mass (and therefore the tension) must be the same in the two cases. Hence then it follows that if A B B'A' represent any actual elevated mass, the direction of the tension at any point c will be that of the tangent line at that point as above described.

There is no difficulty in extending reasoning precisely similar to the above to any more complicated form of the elevated mass, of which the upper and lower surfaces were originally parallel, and horizontal, and we shall arrive at this conclusion.-If we conceive the mass, previous to its elevation, to be composed of horizontal laminæ (or thin strata) the directions of the tensions at any proposed point of the mass when elevated but still unbroken, will lie in the tangent plane to the curved surface formed by that originally horizontal lamina in which the proposed point may be situated; and the intensity of the tensions will be the same, in different lamina at points similarly situated in each.

If the mass in its undisturbed position be not of uniform depth, (i. e. if the upper and lower surfaces be not parallel,) the above reasoning would not be accurately applicable. The case, however, we have considered may be taken as the standard one to which others will approximate with more or less accuracy, particularly as physical reasons might be assigned

There are causes why this should be only very approximately true. (See Memoir, p. 42.)

why an extensive cavity within the earth should be nearly horizontal. Adhering then to this case, it is manifest that the extension of each component lamina of the mass will depend on the form assumed by it when the mass is elevated, since its boundaries, by hypothesis, remain immoveable. Consequently the direction of the tension in the tangent plane before mentioned must also depend upon the form of the lamina. This direction is not generally horizontal, but since it will usually be nearly so, and will always determine the horizontal direction, or azimuth, of a vertical plane drawn through it, we shall be understood when it may be convenient to speak of the horizontal tensions.

It is manifest then that the determinations of the directions of the tangential tensions in the elevated mass, must in cases such as the above be a purely geometrical problem, as may be easily elucidated by a few instances. In the elevation already described (of which the segment of a cylinder, by a plane parallel to its axis, may be regarded as the approximate type, and which may therefore be termed cylindrical) it has been shown that this tension lies entirely in a vertical plane perpendicular to the axis. If the elevation approximate to the form of a cone (which may be conceived to be formed by the superposition of similar conical shells), it may be shown*, that if each lamina remain unbroken, the direction of the only tension will be parallel to the slant side of the cone, and will pass through its axis; but that if a dislocation exist along the vertical axis, the principal tension at any proposed point (particularly near the vertex) will be perpendicular to the vertical plane passing through that point and the axis, there being also another tension in that plane. If again the form of the elevation should approximate to the segment of a sphere, there will be two tensions at each point of the mass, one of which will lie in the plane through the proposed point and the vertical axis of the elevation, the other being perpendicular to that plane.

The above are some of the most simple forms which the elevated mass can be conceived to assume; they may, however, be taken as the approximate types of many of the general elevations which present themselves to our observation, considered independently of their local irregularities. When the superficial boundary of the elevated mass is very irregular, (particularly if the superficial extent be not very great,) the directions of greatest extension, or of greatest tension, will be very different in different points; and it may become very dif

* See Memoir, p. 47.

ficult to calculate with any precision the resulting phænomena. Cases however may easily be conceived without such difficulty, though more complicated than the simple ones above alluded to. Suppose, for instance, recurring to our hypothesis of internal cavities, one cavity of great extent to exist at a certain depth, and another smaller one within the mass above the former, and communicating with it, so that any fluid pressure acting in the lower should be communicated immediately to the upper one. That portion of the elevated mass which lies directly above the upper and smaller cavity, may manifestly be subjected simultaneously to the tension impressed upon the whole mass from the action of the elevatory force in the larger cavity, and to that produced by the partial elevation above the smaller one. These two sets of tensions may be conceived to be superimposed the one on the other, in the same manner as any two sets of forces in equilibrium may be so superimposed *. Their intensities and directions will depend on the forms of the general and partial elevations respectively. Thus we may have a partial elevation of which a cone or segment of a sphere should be the approximate type, superimposed upon a general one of which the type should be the segment of a cylinder. Other combinations might be formed in a similar manner.

Should it appear preferable to consider the subject independently of the hypothesis of internal cavities, we have only to conceive our partial elevations to be produced by a more intense action of the elevatory force at those points. As regards the resulting state of tension, it is perfectly immaterial which hypothesis we adopt.

The states of tension above described refer to the mass in its elevated but unbroken state, i. e. previously to the formation of those fissures which must of course be formed when the tension shall become greater than the cohesive power of the mass. The tension will begin to be produced at the instant the act of elevation commences, and will increase till it acquires the intensity just mentioned. Time will be necessary for this, but it may possibly be so short as to give to the action of the elevatory force the character of an impulsive action, which would probably produce the most irregular phænomena, and such as would be altogether without the sphere of calculation. I exclude therefore the hypothesis of this kind of action, not as involving in itself any manifest improbability, but as inconsistent with the existence of distinct approxima

One of these sets of tensions may possibly modify the other, but in a general explanation, or in a first approximate calculation, this modification may be neglected.

tions to general laws in the resulting phænomena. It would appear probable however that the time above mentioned will be short, and I therefore assume it to be so, and that consequently the tensions increase rapidly but continuously from zero to that degree of intensity which is necessary to overcome the cohesive power of the elevated mass. This assumption has also the advantage of facilitating some parts of the mathematical investigation*.

It will, perhaps, be somewhat more convenient for our further investigations, if we conceive the tensions at different points of one of our elevated, but still continuous and unbroken, component laminæ, transferred to corresponding points of a plane laminat. For this purpose, imagine each point of the curved lamina projected on a plane horizontal one, and that the same tension exists at each point of the latter, as at the point of the former, of which it is the projection; the direction of each tension in the horizontal lamina being the projection upon it of that of the corresponding tension in the curved one. Now one of our ultimate objects will be, to determine the horizontal directions of the fissures which must result in the elevated mass, when the tensions become of sufficient intensity to produce them, and these directions may be considered as coinciding with those which would be produced in our hypothetical horizontal lamina. Consequently our investigation will be reduced to the determination of these latter directions.

To elucidate this, suppose our general elevation to be such as first mentioned above, or what I have termed cylindrical. Its projection on a horizontal plane will be a parallelogram,

D

E

G

F

represented by DEFG. Suppose also a partial elevation

See Memoir, p. 21.

We may remark that the vertical elevation of the disturbed mass, in the state above described, is always extremely small compared with its horizontal extent.

approximately spherical, superimposed upon the general one, such that O shall be the projection of its vertical axis, and the dotted circle that of the circumference of its base. Then taking P as the projection of any proposed point in the partial elevation, we must suppose applied there, first, a tension (F) impressed on the mass generally perpendicular to DE; secondly, a tension (ƒ) in a direction passing through O (see p. 233); and thirdly, another tension f perpendicular to P O. From these data the directions of the fissure through P, when the tensions become sufficient to produce it, must be determined. And here we may remark, that since one lamina of our elevated mass will be similar to another, the tensions F,f and f, will be very approximately the same for each; and that consequently the direction of the fissure just mentioned will equally determine the horizontal direction of the fissure which shall pass through any point of which P is the projection. The extensibility of the mass being assumed to be small, the intensities of the tensions F,ff will be proportional to the extension each would produce in the mass at P, if it acted separately, or to the additional extension produced by each when acting simultaneously. The accurate determination of these intensities would in most cases present great difficulties. In general, however, it will be sufficient to consider such tensions as f, and f (belonging to the partial elevation) merely as forces producing modifications in the effects of F, the nature of which can be determined with sufficient accuracy for practical purposes.

[To be continued.]

XLV. On the Aurora of November 18th, 1835. By the Rev. T. R. ROBINSON, D.D.

To the Editors of the Philosophical Magazine and Journal. GENTLEMEN,

November

MR. R. STURGEON's notice of the aurora of 18th (not 16th as misprinted,) induces me to send you the notes which I made of its appearance, as from his positive statement "that he saw no appearance of aurora to the south of the zenith, though frequently looked for," this seems to be one of the very rare cases where auroral phænomena can be proved to occur in a low region of the atmosphere. They are as follows, the time being reduced to Greenwich.

"Nov. 18. Sky strongly illuminated, but covered with clouds till 9 20m, when two arches were visible, which broke