The

The brightest stars in the nebula are compared in Table V. first and second columns give the Bond number and magnitude. The columns headed A, B, E, and F give the magnitudes derived from those photographs respectively by the method of sequences. The results derived from E and F by the method of Argelander are given in the columns headed E' and F'. The mean values of E and E', and of F and F', are given in Table IV.

One of the most important applications of the determination of photographic magnitudes is to the measurement of the colors of the stars. The rays affecting the photographic plate have in general a less wavelength than those to which the eye is most sensitive. It therefore follows that a reddish star, that is, one in which the rays of great wave-length predominate, will appear relatively too faint in the photograph. The residuals in the last columns of Table IV. will then be positive. A bluish star is similarly indicated by a large negative residual. These residuals form a convenient measure of the color of the stars. In most stars the difference in color is due to slight differences in the relative intensities of the blue and red rays. Until the law defining the relation of the intensity to the wave-length is known, a single number serves to describe the principal cause of the color. Of course in the case of stars in which a large part of the light is concentrated in bands or lines, the residuals will not be directly comparable with those of other stars. Even here, however, this test may be advantageously employed to compare stars of the same class, as, for instance, those of the third type of Secchi.

The first part of Table VI. contains the stars in which the residual equals or exceeds one magnitude. The first three columns give the Bond number and magnitude and the photographic magnitude, taken from the first, fourth, and fifth columns of Table IV. The photographic magnitude was determined a second time to see if the large residual was due to error. The results are given in the fourth column of Table VI. The difference in the two measures is given in the next column, and in the last column the residual found by subtracting the second column from the mean of the third and fourth columns. The second part of Table VI. gives the corresponding values for the blue stars in which the residual has a negative value exceeding one magnitude.

The first part of Table VII. contains the stars given in the Bond Catalogue not contained in the photograph, and accordingly marked d in Table IV. As the faintest stars visible in the photograph have a photographic magnitude of about 15.0, it follows that a slight redness of the stars in Table VII. would account for their absence in the photograph. The stars marked c in Table IV. are Bond 367, 443, 786, and 826; although not visible in F, they were detected in G.

The second part of Table VII. contains the stars which are visible in both the photographs F and G, but are not given in the Bond Catalogue. The successive columns give a current number, the approximate difference in right ascension and declination from 'Orionis, and the photographic magnitude.

Many more objects which cannot be distinguished from stars are visible on either F or G, but not on both. After completing this list, it was compared with the map of the Earl of Rosse (Phil. Trans., 1868, Pl. III.). Stars appear on this map which are moderately near Nos. 4 and 11, but none are near any of the other stars in the second part

of Table VII. None of them are given in the list prepared by Lord Rosse of the stars not contained in the catalogue of Struve (Phil. Trans., 1868, p. 59). A comparison with the map of Mr. Common (Monthly Notices, XLIII. 256) showed that Nos. 10 and 11 were already given there. Mr. Common's stars nf. 690 and np. 750 are not visible on G, although the first of them is well shown on F. The stars near Bond 685 and 741 were not measured on account of the nebulous light with which they are surrounded. Their presence in G is somewhat doubtful. Until the remaining stars are actually seen, we may infer that they are too faint to be visible to the eye, and that our only evidence of their existence is by means of the photographic plate. These stars are also probably of a bluish color. As the number of stars is nearly the same in the two parts of Table VI., we may infer that for white stars the limiting magnitude for the photograph does not differ much from that for the eye.

The agreement of the results given on page 408 is hardly a fair test of the errors of measurement. A better indication is afforded by the repetition of the measurement of the red and blue stars in Table V. The average difference in the results is .14 of a magnitude, which indicates a probable error of each of about .08. The two measures of E by Argelander's method and by sequences give for the 35 stars compared by both methods an average deviation of .20, or a probable error of .12. Forty stars are common to E and F. Omitting the five which are stated on page 413 to be discordant, the average difference in the two magnitudes of the remaining thirty-five is .27. The probable error of each, if they are equal, is .16.

INVESTIGATIONS ON LIGHT AND Heat, made and PUBLISHED WHOLLY OR IN PART WITH APPROPRIATION FROM THE RUMFORD FUND.

XIX.

BRIEF CONTRIBUTIONS FROM THE PHYSICAL LABORATORY OF HARVARD COLLEGE, UNDER THE DIRECTION OF PROFESSOR JOHN TROWBRIDGE.

RELATION BETWEEN SUPERFICIAL ENERGY AND

THERMO-ELECTRICITY.

BY CHARLES BINGHAM PENROSE.

Communicated by Professor Trowbridge, June 11, 1884.

WHEN two media which do not mix are in contact, the particles near the surface have more energy than similar particles in the interior of the media. It is probable that this increase of energy is sensible only within a distance of a thousandth of a millimeter from the surface.

The result of this difference of energy is to render the surface of contact of the media as small as possible. This is seen in Plateau's experiment upon a mixture of oil in alcohol and water.

When the area of the surface is increased in any way the surface energy is increased, and work can be done in the contraction of the surface. The surface behaves exactly as if a tension equal in all directions existed at every point of it. In discussing surface energy it can therefore be considered in two ways. It can be considered as part of the internal energy of the body; and in this case to the internal energy that is generally understood in thermodynamics must be added a term which shall be proportional to the surface, and also a function of the temperature. Or it may be introduced into the equations of the external work: and in increasing the surface it can be said that work is done against superficial tension in the same way that work is done against an external pressure.

The numerical value of the surface energy per unit of area is equal to that of the surface tension per unit of length.

Since the superficial tension depends not only upon the body itself, but also upon the substances in contact with the body, and since work done by, or against, superficial tension depends not merely upon the initial and final states of the body, but upon the manner of passing VOL. XX. (N. S. XII.)

27

from one state to the other, it is probably best to introduce surface energy into the equations of the external work.

Let the state of any body be represented as a function of the two independent variables x and T,- where 7 is the absolute temperature. If the body is an homogeneous solid, its mechanical state is represented as a function of six independent variables. The equations, however, deduced by considering only the variations of x and T can be extended to the case of a solid by adding five other equations of the same form.

When the body undergoes any small transformation, the quantity of heat absorbed expressed in mechanical equivalents - is:

The coefficient "a" depends upon the external and the internal works. It is the mechanical equivalent of the latent heat relative to the vari

b

able x. represents the specific heat of the body for a constant meJ chanical state.

The external work is done against superficial tension and external pressure: hence "a" depends upon these two quantities.

The value of the external work in the above transformation is:

S is a function of the superficial tension and of the shape of the body. The work done against superficial tension is equal to the coefficient of capillarity or the coefficient of superficial energy into the increment of area. the coefficient of capillarity, and let

Let

where A equals the area of the surface of the body.

be substituted

S= k f(x).

Then for S may

Let Frepresent the total energy of the body:

d Fadx+bd T+pdx+ Sdx.

For any closed cycle the total variation of F is zero.

III.

The value

of an increment of F depends merely upon the initial and final states of the body. d F is therefore an exact differential.