n : D (3) Bm=9m—1=5X1; for a star of the nth magnitude =(4) BM = 01-1=SX1: also ' Bm (5) в, = m -1 = SXiXD* (1)"-" SXi XD" If m and n are known on some scale whose ò is known, or if Bm Br 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 i=i' then these equations become (6) B'= 4= R'?Xi 4R ; (7) B" Xi R'? XD"? R' XD There are three special cases which we may examine: I. Stars of known distances D', D", etc., or of known parallaxes a', 7", 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. ..2 ; and 2 I. B In the case of stars of known parallaxes, a', 7", etc., the equation (8) becomes R:?X? R"X" Table A (page 140), contains the result of recent measures of well determined parallaxes (excluding double stars). We may assume oc Lyrve as the unit star, so that B'' = 1 and R =1 and deduce R for each star ; i. e., the diameter of each star relative to the diameter of oc Lyrae. 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 i=i=i", etc., is erroneous. We may go farther and apply the formulæ to double stars and clusters. R' 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 0.53 TABLE A. Table II contained forty stars certainly binary, the components being of different colors : the mean difference of magnitudes (B-A) is 2.44". 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.” Our equation (8) is sm-n= R''X D'' R"? XD2 For any pair of stars, certainly binary, D'=D" and the ratio can be taken out of the following table, which is computed 2 2 R'? R "SEIDEL: Ueber d. gegenseitigen Helligkeiten der Fixsterne (1852). The other numbers in this column are interpolated with 8=0.40. with various values of 0 (the light-ratio), by entering it with 12 — m, i. e., the difference in magnitude of the component stars. n -- m TABLE B. GIVING TIIE VALUES OF ()nm FOR TIIE LIGHT-RATIOS OF ARGEL ANDER, STRUVE AND J. HERSCHEL.5 0.9 1.000 1.10 1.20 1.32 1.14 1.582 1.73 1.90 2.07 2.27 2.500 2.74 3.00 3.29 3.61 3.353 4.33 4.74 5.21 5.71 6.250 6.85 7.52 8.26 9.09 10.000 10.75 11.79 12.94 14.16 15.625 17.1 18.7 20.5 1.000 1.11 1.23 1.37 1.52 1.695 1.88 2.09 2.33 2.58 2.857 3.17 3.52 3.91 4.35 4.831 5.38 5.99 6.67 7.41 8.130 9.01 10.00 11.11 12.35 13.717 15.24 16.95 18.83 20.92 23.310 25.8 28.6 31.7 35.2 39.06 43.5 48.3 53.8 59.9 61.67 74.1 82.0 90.9 101.0 112.23 121.7 138.5 153.9 171.2 190.48 1.000 1.08 1.16 1.25 1.34 1.445 1.56 1.68 1.80 1.94 2.083 2.25 2.40 2.59 2.79 3.003 3.24 3.48 3.75 4.03 4.348 4.67 5.03 5.41 5.81 6.250 6.71 7.25 7.81 8.40 9.091 9.7 10.5 11.3 12.1 13.06 14.0 15.1 16.3 17.5 18.80 20.2 21.8 23.5 25.3 27.17 29.2 31.5 33.9 36.5 39.06 20.57 27.0 29,6 32.5 35.6 39.06 42.9 46.7 51.3 56.2 61.73 67.6 74.1 81.3 89.3 97.66 * This table is the conrerse of that given by Johnson in the Radcliffe Observations, Fol. XII , p. 31, Appendix, which is computed with three significant figures. If the underlying hypothesis i=i', etc., is true, it would follow that of the stars of Table I the larger is on the average 1.6 times the surface of the smaller, while Table II would show that on the average the surface of the larger star of each pair was 9.1 times that of the smaller for pairs of different colors. It may be said in passing that the colors of the small stars in Table II are not due to their proximity to the large stars, but are intrinsic. The doubt in the conclusion as to relative size is a doubt whether the fundamental hypothesis of cqual intrinsic brilliancy is true. Still keeping to this assumption we can go farther, as Prof. PICKERING has shown (op. cit., pp. 6, 7), for we can find the mass of each component of a binary star of known parallax. Thus if M and M' be these masses, a the mean distance of M and M' and T the periodic time in years D and p being the distance and parallax of the binary D=1; sin p. If an is its semi-major axis in seconds Thus M + M the unit of M and M' being the sun's mass. Knowing w, a" and T for any binary, we may deduce the dimensions of each component. Thus for of Centauri, p = 0”.9 (GYLDÉN), a'' = 155.5, T=80.0 years (Russell in Monthly Notices, R. A. S., 1877), whence M + M'= 0.80 of the sun's mass. Sir Joun Ierschel in the Cape of Good Ilope observations (p. 300), says the components of a Centauri are of the same color and <iffer 0.73 magnitucles (A = 1.00, B= 1.73), o for o IIERSCIIEL is 0.480 wlience R* ? = (2.083) 0.13 = 1.71, R M These results differ from those given by Prof. PICKERING (page 6), who makes = 63.0 on account of a different assumption as to the relative magnitude of the two components. The recent research of Dr. ElKIN, Ueber die parallaxe von o Centauri leads to the conclusion (p. 25), that the masses M and M' are equal. III. Let us apply this hypothesis to the case of a cluster of stars. The cluster contains say N, stars all at a distance D, from us. Suppose the cluster to be 10' in diameter. Beside the N, stars which really make up the cluster, others are seen in the same field of view: that is the cluster, as we see it, consists of two classes of stars. First, the N, stars really associated with each other in a physical connection and at the same distance D, from the sun, and second, those scattered on the background of the learens which (accidentally) are seen among the component stars of the cluster. The last may be practically eliminated from the cluster by examining many fields of view 10' in extent situated about the cluster in question. The average number and magnitude of the stars in such 10' fields can be determined and subtracted from the stars of the real cluster. Then there remain N, stars of a magnitude from a to b for example. At the unknown distance D, of this particular cluster we learn that stars of the apparent magnitudes from a to 6 are interinixed. From other clusters we may also determine like data. But all the stars of the real cluster are at (essentially) the same distance D, and assuming them to have the same power to radiate light per unit of surface the range of magnitudes (a .....b) will afford us a means of determining the probable range of size R, .... R, of the component stars. For a star of the brightest magnitude a, gives out an amount of light expressed by sa-1, and similarly the faintest star of the cluster gives out the light ô6-1. As these stars are at the same distance their relative brightness is proportional to ja-v= * ?Xi'. Thus for this particular cluster the assumption i' = i" gives us the relation R' = R' X 7". Other clusters will give us R"=Riix 7" etc., and by collecting all the cases where X 82 |