where we employ the symmetrical notation of Todhunter and Pearson's History,' and refer everything to fixed Cartesian axes. Also if X', u', v denote the direction-cosines of the outward-drawn normal at any point on a surface, and F, G, H be the components of the applied force there per unit area of surface, we must have

where p is any constant.

If the material be bounded by one surface Se, this is obviously the solution for a uniform pressure p over that surface, for the components of p are

so that the equations (7) are satisfied.

Answering to (8) we have everywhere a uniform "dilata

tion" ▲ given by

where k is the bulk-modulus *.

e

As A is uniform, the reduction SV, of the volume V, enclosed by S. is given by

Suppose, now, any imaginary surface S; drawn in the material enclosing a volume V. Then since (8) holds everywhere, it holds over the surface S, and so glancing at (7) we recognize that the stress over S; is a uniform normal pres

*See Thomson and Tait's 'Natural Philosophy,' Part ii. art. 682, and Love's 'Elasticity,' vol. i. art. 41,

sure p. Thus in a shell bounded by S; and S, under equal uniform pressures p on the two surfaces, the stresses, and so the strains and displacements, are at every point the same as if S were an imaginary surface drawn in material completely filling Se, and p were applied over Se only. In particular, the changes in the volumes contained by the surfaces S, and Se must be the same in the two cases.

i

e

In the case of the shell, let &V and SV be the increases in the volumes V, and V, when uniform pressure p is applied over the inner surface only, and let &V and SV" be the corresponding reductions when the same pressure p is applied over the outer surface only.

i

i

e

In the case of the solid, bounded by the one surface Se, let SV; be the reduction in the partial volume V1, and SV. the reduction in the total volume V, due to uniform pressure p over Se. Then by what has preceded, since stresses are superposable,

8V!+(—¿V'})=8V¿,

SV"+(—¿V'%)=8Ve.

But in the case of the complete solid ▲ is uniform, being given by (10), and so

or, after Dr. Guillaume's definition, see (1) and (2),

(14)

(15)

the result required.

Again substituting for dV, from (14) in (13), we get

e

Dr. Guillaume had no occasion to arrive at this latter result. To put it in a similar form to the other, let y; represent the increase per unit volume of V, due to unit pressure per unit surface of Si, and let ye represent the reduction per

unit volume of V, due to unit pressure per unit surface of Se,

then

e

Since the result (3) may come under the notice of some persons whose interest in thermometry is not accompanied by a knowledge of elastic solids, it may be well to state explicitly under what conditions it has been proved to hold, and wherein these conditions may differ from what occurs in practice with actual thermometers.

The conditions assumed in the mathematical theory are:(A) that the material is completely homogeneous, though not necessarily isotropic;

(B) that the pressure is perfectly uniform over the surface where it is applied;

(C) that the volume whose change is considered is the entire volume within the inner surface.

The fact that (3) holds when all these conditions are satisfied does not of course necessarily imply that it ceases to hold if one of the conditions is not satisfied. As a matter of fact it certainly holds in one case when (A) is only partially satisfied, viz. in the case of a spherical shell composed of concentric layers of different isotropic materials, which though differing in rigidity have all the same compressibility. But it is obvious that in any case where the compressibility, and so the bulk-modulus, is not uniform it would be meaningless. There is room for doubt as to how far condition (A) is satisfied by thermometers. Differences of elastic quality between the bulb and stem, or even between the material at the outside and inside of the stem, seem not unlikely to occur.

The condition (B) is probably never satisfied exactly; and it may be very far from holding when the stem is vertical, in the case either of internal pressure or external fluid pressure. It may, however, be regarded as practically satisfied in the case of external atmospheric pressure.

The preceding mathematical theory gives no direct and certain information as to how the change of volume is divided between the bulb and stem even when the pressure is uniform. If, however, we for a moment supposed the bulb and a short

adjacent portion of the stem to be converted into a closed vessel by means of a flat disk of glass closing the bore, the effect on the change of the volume so enclosed could hardly differ appreciably from that occurring previous to the closure, supposing the bore to be fine. Also if the bulb be nearly spherical the pressure over its surface would seem, so far as change of volume is concerned, to be replaceable without serious error by a uniform pressure equal to that actually found at the level of the bulb's centre.

We should thus conclude that the change in the volume of a nearly spherical bulb follows approximately the same law as if the bulb were closed and subjected to uniform pressure equal in intensity to that occurring at the level of its centre of gravity. A relation equivalent to (3) thus seems likely to hold approximately for the bulb alone, at least when it is nearly spherical and the bore is fine. This is, I think, practically in harmony with the conclusion reached by Dr. Guillaume on his p. 111.

XLIII. On a New Method for Mapping the Spectra of Metals. By Prof. HENRY CREW and Mr. ROBERT TATNALL*.

HE difference in physical character between the various

lines in the spectrum of an element has recently assumed such importance that a table of wave-lengths is now, to some extent, incomplete unless accompanied by a photographic map. This is especially true for one who is seeking new relations among the wave-lengths. Thus, in the case of cadmium, the triplets overlap, but, "owing to the physical similarity of the lines forming any one triplet, it is a matter of perfect ease to select them "

't.

Indeed, in many cases where series have been discovered, one might decide to what series a given line belongs quite as well by its appearance as by its wave-length. Rydberg has happily suggested, for these series, names which describe the appearance of their respective lines.

So far as we are aware, all photographs of metallic spectra which have hitherto been made are, with two exceptions, either of spark spectra or spectra of substances vaporized in the carbon arc. The two exceptions to which we refer are, first, the well-known spectrum of iron by Kayser and Runge, in which the arc employed is that between iron rods about one centim. in diameter; and, secondly, a copper arc with *Communicated by the Authors. Ames, Phil. Mag. July 1890, p. 45.

which these same gentlemen have attempted to vaporize strontium, and thus obtain the strontium triplet* at X3800, free from the cyanogen band. They say, however, that the arc worked so badly as to give only one line out of the three.

The well-known difficulty with the spark spectrum is that it is almost as characteristic of the slight differences in physical condition under which it is obtained as of the chemical element from which it is obtained. Not only so, but owing to its streaks, as it were, of high temperature ("luminescence"?) there is obtained, at the same time with the spectrum of the metal, also the spectra of the gases in which the discharge takes place.

In the case of the carbon arc, nature has fortunately grouped its many thousand lines into bands, leaving here and there comparatively clear spaces in which the lines due to substances deliberately introduced into the arc can be studied and measured with a high degree of accuracy, as exemplified in the work of Rowland and of Kayser and Runge.

Fortunately also, in the case of some metals, especially the easily volatile ones, the metallic vapour acts as if it shunted off the current from the carbon vapour; and the metal comes out strong in comparison with the carbon.

At the same time, the carbon and cyanogen bands stretch practically through the whole spectrum from λ 3500 into the infra-red. Not only so, but many of these carbon lines have, as a rule, intensities quite comparable to those of the metallic lines. One ingenious effort has been made by Kayser and Runge (1. c.) to rid themselves of the cyanogen bands by working the carbon arc in a current of carbon dioxide. This is partially successful; but, at best, it only diminished the intensity of the band. Messrs. Lewis and Ferry †, speaking of the infra-red spectra of the metals, say:-" It seems as though little more could be done in the discovery of new metallic lines unless the carbon lines are first carefully mapped, or some means is devised for raising the substances investigated to sufficiently high temperature without placing them directly in the [carbon] arc."

We have, therefore, devised and used during the past year the following method for obtaining the arc spectrum of the metallic elements free from carbon, free from air-lines, and free also from any continuous spectrum.

The idea is simply that of an arc in which one pole rapidly

*Kayser and Runge, Wied. Ann. lii. p. 115 (1894).

† Johns Hopkins University Circular, May 1894.