theory. Indeed, it is greatly due to the stimulus of his appreciation that the telephone became an accomplished fact.

I cannot state too highly also the advantage I derived in preliminary experiments on sound vibrations in this building from Professor Cross, and near here from my valued friend Dr. Clarence J. Blake. When the public were incredulous of the possibility of electrical speech, the American Academy of Arts and Sciences, the Philosophical Society of Washington, and the Essex Institute of Salem, recognized the reality of the results and honored me by their congratulations. The public interest, I think, was first awakened by the judgment of the very eminent scientific men before whom the telephone was exhibited in Philadelphia, and by the address of Sir William Thomson before the British Association for the Advancement of Science. At a later period, when even practical telegraphers considered the telephone as a mere toy, sev eral scientific gentlemen, Professor John Pierce, Professor Eli W. Blake, Dr. Channing, Mr. Clark, and Mr. Jones of Providence, R. I., devoted themselves to a series of experiments for the purpose of assisting me in making the telephone of practical utility; and they communicated to me, from time to time, the results of their experiments with a kindness and generosity I can never forget. It is not only pleasant to remember these things and to speak of them, but it is a duty to repeat them, as they give a practical refutation to the often repeated stories of the blindness of scientific men to unaccredited novelties, and of their jealousy of unknown inventors who dare to enter the charmed circle of science.

I trust that the scientific favor which was so readily accorded to the Telephone may be extended by you to this new claimant,"The Photophone."

ON SOME OF THE CONSEQUENCES OF THE HYPOTHESIS, RECENTLY PROPOSED, THAT THE INTRINSIC BRILLIANCY OF THE FIXED STARS IS THE SAME FOR EACH STAR. By EDWARD S. HOLDEN, of Washington, D. C.

I.

In all statistical researches upon the arrangement of the fixed stars in space, it has been found to be necessary to make some fundamental assumption, more or less probable.

Some assumption is forced upon us by our complete ignorance of the nature of the stars themselves and of the real laws according to which they are distributed.

The fundamental assumptions have usually been, either that the stars are all of equal brightness; or else that they are equally scattered, so that within equal portions of space, equal numbers of stars exist. Various modifications of these two hypotheses have been made and their consequences worked out; but some form of one of them has usually been the starting point.

It may be of interest to see the consequences which follow from an assumption less violent than either of the preceding. This is that the brightness of the unit-area of all stars is the same. The simple formulæ which relate to this subject were put in a form which allows the consequences of each of the three fundamental assumptions to be seen, in NEWCOMв and HOLDEN'S Astronomy,2 page 489. The hypothesis that the brightness of the unit-area of all stars is the same has been recently made the basis of computation in a paper by Prof. E. C. PICKERING, "Dimensions of the Fixed Stars, etc., reprinted from the Proceedings of the American Academy, Cambridge, 1880," in which (page 35) it is used to determine the dimensions of a dark satellite to Algol. As we can have no à priori proof of the truth of this hypothesis, it will perhaps be useful to trace its consequences in various directions. Although the data at our command are not sufficient to enable us to come to certain conclusions, yet they may be sufficient to test the value of the fundamental assumption, and it is only for this reason that I bring them together. It may not be improper to say

1 Dimensions of the fixed stars, etc., by Prof. E. C. PICKERING, Cambridge, 1880, p.3. 'American Science Series, New York, 1879.

that they were deduced in 1877, and would have remained unpublished except for their bearing on the present question.

While the first two assumptions in regard to the distribution of stars have been shown to be roughly and in a general way approximations to the truth 3 we know that they are in fact untrue. For the hypothesis that all stars are of equal brightness, and thus that stellar magnitude depends on stellar distance alone, is contradicted directly by the determinations of parallax, and more glaringly and in a more general manner by the existence of clusters, in which stars of different brightness are associated at the same distance from the earth.

The hypothesis of equable distribution is negatived by the existence of clusters at all, so that while this supposition is also in a general way true, it needs serious modifications to make it fit special cases. The third hypothesis of equal brightness of the unit-area of the surfaces of all stars is certainly not true in every case, but in any case it is less violent than either of the others à priori. The objections to it we may consider later. It may be mentioned here, that of the 324,000 stars from first to ninth magnitude, we know considerably less than 1,000 highly colored stars; the vast majority of stars being white. This is in no sense a proof of the assumption. It does not militate against it, however, and is what might be expected if the assumption were indeed true. We may express our conditions in an algebraic form as follows: If S be the surface and R the radius of a star at distance D; i the amount of light emitted per unit of surface, Bm, its brightness in arbitrary units as seen from the earth, m, its stellar magnitude on any scale whose light-ratio is 6, then the most general expression for Bm is

(1)

Bm=X if light is not extinguished in space, or

D

The unit of D is arbitrary.

If the brightness of an average first magnitude star is unity, B1 = 1,

then

(2) Bm-im

; so that for a star of the mth magnitude

3 Not only by the results of W. HERSCHEL, but by the later researches of PETERS, GYLDEN, C. S. PEIRCE, GOULD, and others.

(3) Bm=¿m-1=5X; for a star of the nth magnitude

If m and n are known on some scale whose is known, or if Bm Bn are measured photometrically, then for each star one of the quantities S, i, D, can be expressed in terms of the other two, and in general this is all that can be done. If we assume throughout ii' then these equations become

There are three special cases which we may examine: I. Stars of known distances D', D", etc., or of known parallaxes ', ', etc. II. Binary stars where D' D", although both D' and D" are unknown in general. III. Clusters, where D'D", etc., and D', D', etc., are unknown.

I.

In the case of stars of known parallaxes, ', ", etc., the equation (8) becomes

Table A (page 140), contains the result of recent measures of well determined parallaxes (excluding double stars).

We may assume Lyre as the unit star, so that B" 1 and

R'
R"

R1 and deduce for each star; i. e., the diameter of each star relative to the diameter of Lyra. The table shows the largest diameter to be 271 times the smallest, or not far from the ratio of the Sun's diameter to Mercury's (291 to 1). This difference corresponds to an immense difference of mass, but perhaps not sufficient to show that the fundamental hypothesis ii'i", etc., is erroneous. We may go farther and apply the formula to

double stars and clusters.

II.

In a note in the American Journal of Science for 1880, June (page 467), I gave certain tables of binary stars which I had prepared in 1877 with reference to this subject.

Table I, there given, contained 122 stars certainly binary and with component stars of like color. The magnitudes and colors are from the best authorities. The mean difference of magnitudes (B-A) is 0.53m.

Table II contained forty stars certainly binary, the components being of different colors: the mean difference of magnitudes (B-A) is 2.44m.

These tables showed that considering every known case of binary stars of known color: I. The components of the 122 binary stars of the same color differ in magnitude on the average only 0.5m.

II. The components of the 40 binary stars of different colors differ in magnitude on the average 2.4m."

R. 2

R2

For any pair of stars, certainly binary, D'D" and the ratio can be taken out of the following table, which is computed

SEIDEL: Ueber d. gegenseitigen Helligkeiten der Fixsterne (1852). The other numbers in this column are interpolated with &=0.40.