unity, he computes the superior limit of density for 12 variables, based on their periods and the duration of their partial eclipses. The greatest limit is in the case of Z Herculis and is 0.728. The least is in the ease of S Caneri and is 0.035. The average is about 0.2. As the actual density may be less than the limit by an indefinite amount, the general conclusion from his work may be regarded as the same with that from the work of Roberts.
The results of the preceding theory are independent of the parallax of the stars. They, therefore, give us no knowledge as to the mass of a binary system. To determine this we must know its parallax, from which we can determine the actual dimensions of the orbit when its apparent dimensions are known. Then the formula already given will give the actual mass of the system in terms of the Sun's mass.
There are only six binary systems of which both the orbit and the parallax are known. These are shown in the table below. Here the first two columns after the stars named give the semi-major axis of the orbit and the measured parallax. The quotient of the first number by the second gives the actual mean radius of the orbits in terms of the earth's distance from the Sun as unity. This is given in the third column, after which follow the period and the resulting combined mass of the system. The last column shows the actual amount of light emitted by the system, compared with that of the Sun.
| a." | Par. | a. | Period. | Mass. | Light. | |||||||
| " | " | y. | ||||||||||
| η Cassiopiæ | 8. | 21 | 0. | 20 | 41. | 0 | 195. | 8 | 1. | 8 | 1. | 0 |
| Sirius | 8. | 03 | 0. | 37 | 21. | 7 | 52. | 2 | 3. | 7 | 32. | 0 |
| Procyon | 3. | 00 | 0. | 30 | 10. | 0 | 40. | 0 | 0. | 6 | 8. | 5 |
| α Centauri | 17. | 70 | 0. | 75 | 23. | 6 | 81. | 1 | 2. | 0 | 1. | 7 |
| 70 Ophiuchi | 4. | 55 | 0. | 19 | 24. | 0 | 88. | 4 | 1. | 8 | 0. | 7 |
| 85 Pegasi | 0. | 89 | 0. | 05 | 17. | 8 | 24. | 0 | 9. | 0 | 2. | 2 |
Even in these few cases some of the numbers on which the result depends are extremely uncertain. In the case of Procyon, the radius of the orbit, can be only a rough estimate. In the case of 85 Pegasi the parallax is uncertain. In the case of η Cassiopiæ the elements are still doubtful.
So far as we have set forth the principles involved in the question, we do not get separate results for the mass of each body. The latter can be determined only by meridian observations, showing the motion of the brighter star around the common center of gravity of the two. This result has thus far been worked out with an approximation to exactness only in the cases of Sirius and Procyon. For these systems we have the following masses of the companions of these bodies in terms of the Sun's mass:
