10 



PROCEEDINTxS OP THE AMERICAN ACADEMY 



telescope not capable of separating them. Let I denote the light of 

 the fainter star in terras of the brighter, and m the magnitude of the 

 fainter minus the magnitude of the brighter. Then, on Pogsou's sys- 

 tem, m = — 2.5 log I. If Mis the magnitude of the brighter star minus 

 that of a star equivalent to the two combined, or having the light (I -\- l), 

 then M^ — 2.5 log (1 -j-Z). From these formuls we can always 

 find the corresponding values of J/ and m. The maximum value of 

 Jf == 0.75 when ?« is zero or the stars are equal. Table IV. enables us 

 to determine M to the nearest tenth of a magnitude for any value of m. 

 As an example, suppose two stars have magnitudes 2.0 and 3.0; then 

 m = 3.0 — 2.0 = 1.0, and M, from the table, lies between 0.35 and 

 0.45 or equals 0.4. The light of both combined will therefore equal 

 2.0 — 0.4= 1.6. 



TABLE IV. — Combination of Two Stars. 



It is sometimes convenient to know what would be the magnitude 

 of a star whose mass was equal to that of the two components of a 

 double star of the same density and brightness. Let m' equal the 

 difference in magnitudes of the two components, and / and n, the light 

 and mass of the fainter in terms of the brighter. Then 



m' = — 2.5 log I = — 2.5 log n^ = — 1.G7 log n, 



since the light is proportional to the square, and the mass to the 

 cube, of the diameter. If then 31 equals the magnitude of the 

 brighter component minus that of both combined, we shall have 

 AI = 1.67 log (1 -|- n), from which 31 is determined as before from 

 any given value of ?«'. The third column of Table IV. gives the 

 value of m' corresponding to every odd twentieth of a magnitude of 31. 

 The value of the latter may thus always be determined to the nearest 

 tenth of a mngiiitude. The maximum value of 31 is 0.50, when 

 m' = 0. Adopting the same magnitudes as in the last example, if two 

 stars have the magnitudes of 2.0 and 3.0, m' will equal 1.0. This 

 value from the third column of Table IV. will correspond to a value 

 of JI/ lying between 0.15 and 0.25, or will equal 0.2. The magnitude 



