and by direct measurement we find that

whence, from (6),

E(e, 4)+ E(e, 4,)=1·037 ;

e2(0-0) cot a=1.139

(the coefficient on the right-hand side becoming 1+). The second side of (1) becomes now 1.184. This value does not agree sufficiently with 1.071 for the difference to be due to any errors of observation; and it follows that Ammonites planorbis was produced from a more involute shell than that called Ammonites erugatus: and such more involute varieties occur in the south of England.

19. To find the angle of elevation in a turbinated shell.

The point in the tracing-curve whose angle of elevation is measured must, of course, depend on the shape of that curve. When it is such as to form a complete cone, as in Eulima, the angle for any point on the surface is directly measurable. When the curve is such as the ellipse, the angle measured by the tangent lines does not correspond to the y already used, but is connected with it by an easily expressed relation; for if o be the semi- vertical angle of the tangent cone, the equation (A) must give equal roots when rcoto is written for z, which requires that

W

tan y=cot w+σ√x2(cot a cote + 1)2 + (cot w-cot e)2. When, as is often the case, we, this reduces to

It is obvious that when w and y can both be observed directly, this gives an equation to find o, as noted in § 15.

An example of the use of this equation may be taken from the previously quoted Cyclostoma elegans. Here

whence

σ='46, w=24°, e=156°, x=87315,

the angle derived from direct measurement being 7810. 20. The semi- vertical angle of the cone for any particular point may also be calculated by means of other angles often more easily measured. Let w be such an angle, and i the angle between the tangent at that point and the axis. Let AB (80) be an element of the curve at that point, DB its projection on the plane of xy (ds). Then

dz do cos i, ds du sini, dz=dr cot w, dr=ds cos a; dz=do

ds=do

If x be the angle between the tangent and the radius from the vertex,

dz sec w do cos x;

=

.. cos w= cos i secx.

(2)

The values, therefore, of i and x are sufficient to give w and . This last equation was obtained by Professor Moseley by a different method, and the first, under a different form, deduced from it. It follows from (1) that i increases with ∞ in @ the same shell, and hence that the angles must be measured at corresponding points. In the same way x is dependent on w, since from (1) and (2) we may deduce

tan x= tan a sin w.

(3)

The angle x is the only one which can be well measured on specimens, and i only on figures. The "sutural angle" employed by D'Orbigny is different from either of these, being the angle between the line joining the ends of two radii corresponding to and 0+ with the smaller of them. The connexion of this with the drawing of the suture is only approximate, as the latter does not lie in one plane. If be this angle, we have

or

If w be found from the tangent lines to the shell, must be measured from the points of contact and not in the suturewhence the term "sutural" angle is inexact for a second reason. This method has the advantage, however, of being applicable to very little more than a complete whorl, the rest requiring one and a half. When less than one whorl is preserved, it is sometimes possible to measure the angle between the projections of two tangents to the curve at points separated by half a whorl, on a plane through the axis perpendicular to that containing them, i. e. between BG (fig. 10) and the corresponding line on the opposite side. If 24 be this angle, we have

One important value of these equations is to serve as a check on the separate measurements, and thus to gain obser

vations for drawing an average, as there is no doubt that spiral shells are not absolutely geometrically constant, and all measures are therefore more or less approximate.

ω

One example of their use will suffice: this may be Phasianella striata. In this we have w=16°; i by observation is 79°. Also R=1.2; to this value corresponds in the Table 86° 41′. Substituting these values of w and a in (1) we obtain i=78° 41′. Again, substituting in (4) for R and w, we obtain =91° 36′; by observation the same angle is 92°. These results are sufficiently close to prove the regularity of the shell formation in this case.

It has not been thought worth while to study the effects of compression on turbinated shells, because from their shape no assumption can be made as to the direction of pressure that could be considered generally applicable.

XXXIII. On the Limits of Hypotheses regarding the Properties of the Matter composing the Interior of the Earth. By HENRY HENNESSY, F.R.S., Professor of Applied Mathematics in the Royal College of Science for Ireland*.

FROM

1. ROM direct observation we are able to obtain only a very moderate knowledge of the materials existing below the solid crust of the earth. The depth to which we can penetrate by mining and boring operations into this crust is comparatively insignificant; and these operations give us little knowledge of the earth's interior in comparison with what is afforded by the outpourings of volcanos. Two hundred active volcanos are said to still exist, while geologists have established that many thousands of such deep apertures in the earth's crust have existed during remote epochs of its physical history. The source or sources of supply for all these volcanoes have poured out a predominating mass of matter in a state of liquidity from fusion. Evidence is thus furnished that matter in a state of fluidity exists very widely distributed through the earth. The supposition that this fluid fills the whole interior, and that the solid crust is a mere exterior envelope, is usually designated as the hypothesis of internal fluidity. From this hypothesis mechanical and physical results of primary importance in terrestrial physics may be deduced.

Newton, Clairaut, Laplace, Airy, and other illustrious mathematicians have used an extension of this hypothesis in discussing the earth's figure. They supposed the particles com

* Communicated by the Author, having been read before the Mathematical and Physical Section of the British Association for the Advancement of Science, Dublin, August 1878.

posing the earth to retain the same positions after solidification as that which they held before it. I ventured, for the first time, to discard the latter portion of the hypothesis as useless and contrary to physical laws. I now venture to say that, in framing any hypotheses as to the physical character of the matter of the earth, we should not affix any property to the supposed matter which is opposed to the properties observed in similar kinds of matter coming under our direct observation. Observation has disclosed that liquids are in general viscid, and that they possess what has been designated internal friction in a high degree*. Observation has recently shown that among the three states of matter (gaseous, liquid, and solid) a law of continuity exists. Observation also discloses that gases and vapours are, of all forms of matter, the most compressible, that liquids are much less compressible, and that solids are still less compressible. Thus, for instance, water is about fourteen times more compressible than copper or brass.

2. If these general comparative properties of liquids and solids are admitted, it follows that in the hypotheses regarding the earth's internal structure we should most carefully guard against any assumption directly in contradiction to such properties. By assuming that the earth contained a fluid totally devoid of viscidity and internal friction, the late Mr. Hopkins attempted to prove the earth's entire solidity. He only proved that it did not contain any of this imaginary fluid; but he by no means proved the non-existence of a liquid possessing the properties of viscidity and internal friction common to all liquids. In the Comptes Rendus of the Academy of Sciences of Paris for 1871 is a paper in which I have given a résumé of the arguments against Mr. Hopkins's conclusions as to the earth's complete solidity; and in the subsequent discussions my priority on this matter seems to have been fairly and honourably acknowledged t. In a recent admirable work on Geology, Pfaff's Grundriss der Geologie, the author gives a brief account of the bearing of astronomical and mathematical investigations on the internal structure of the earth; and he very justly says that the results of observation compel us to regard the earth as for the most part fluid, in order to bring these results into harmony with calculation. Professor Pfaff

As having a special connexion with this subject, see a Report by the Author on Experiments on the influence of the molecular condition of fluids on their motion when in rotation and in contact with solids (Proceedings of the Royal Irish Academy, 2nd series, vol. iii. p. 55).

"Remarques à propos d'une Communication de M. Delaunay sur les résultats fournis par l'Astronomie concernant l'épaissseur de la croûte solide du Globe," Comptes Rendus de l'Inst. France, Mars 6, 1871,

p. 250.

attributes this conclusion to Hopkins, whereas it is precisely that which I had long since enunciated, and is entirely opposed to the views of Mr. Hopkins. More recently Sir William Thomson and Mr. Darwin have investigated the tidal action of an internal fluid nucleus upon its containing solid shell. They have both supposed the liquid to be totally incompressible, and the containing vessel to be elastic and therefore compressible. They have thus given the liquid a property which no liquid in existence possesses, and the solid a property which solids possess in a much less degree than liquids. Their hypothesis is thus totally inadmissible as a part of the problem of inquiry into the earth's structure. I at once admit that a thin elastic spheroidal envelope filled with incompressible liquid and subjected to the attractions of exterior bodies would present periodical deformations, owing to tidal action far surpassing the tides of the ocean. But I do not admit that such impossible substances can represent the materials of the earth. My hypothesis is that the liquid interior matter, instead of being incompressible, is, like all liquids we observe, relatively far more compressible than its solid envelope. A highly compressible liquid contained in a very much less-compressible shell would be a hypothesis more in harmony with physical observation. The tidal phenomena of a compressible fluid, it is easy to see, would be very different from those of an incompressible fluid. The work done by the action of certain disturbing bodies in the strata of compressible fluid would partly result in causing variations of density, instead of producing tidal waves of great magnitude. This has been already shown in the Mécanique Céleste by Laplace, in discussing the tides of the atmosphere. Theory shows that the atmospheric tides should be nearly insensible, notwithstanding the great depth of the atmospheric column, because the work done in the atmosphere is very different from what is peformed in the less-compressible water of the ocean. Observation has fully verified this result.

3. It is admitted that the earth's density increases from its surface towards its centre. If its interior is occupied by a compressible fluid, the law of density of this fluid would result from the compression of its own strata; just as the law of density of the atmosphere is produced by the pressure of the upper atmospheric layers upon those below. But instead of supposing the interior of the earth to be filled by a fluid thus conforming to the observed properties of fluids, both Sir William Thomson and Mr. Darwin have applied their great powers as accomplished mathematicians to the tides of an incompressible and homogeneous spheroid, such as I admit to have no real existence whatsoever.