210 



KNOWLEDGE 



[No^-EMBER 2, 1891. 



parallax, the Suu would be reduced to 5-0 magnitude. 

 This would make 70 Ophiuchi about 2-27 times the bright- 

 ness of the Sun. Accordint; to Dembowsld there is a 

 difference of 1 -7 magnitude between the components. If 

 we assume that each has the same density as the Sun, I 

 find that the combined mass of the two stars would be 

 2-82.5 times the solar mass, which agrees closely with the 

 result found fi-om the orbit. We may, therefore, conclude 

 that Kriiger's parallax for this star is not far from the truth. 

 The diameters of the components would be about l,188,OU0 

 miles, and 5-12,000 miles, and the distance between them 

 27'77 times the Sun's distance from the earth, or some- 

 what less than the distance of Neptune from the Sun. 



7. 85 Pegasi. — For this binary pair, a somewhat 

 doubtfvd paraUax of 0-054". found by Briinnow, combined 

 with Schaeberle's elements of the orbit (P=22-3 years, 

 «=0-9G"), gives a mass of 11-3 times the Sun's mass. 

 Placed at the distance indicated, the Sun would be reduced 

 to a star of 7-41 magnitude. 85 Pegasi was measured 

 5-83 magnitude with the photometer at Harvard, so that 

 the star is 1-58 magnitude, or 4-286 times brighter than 

 the Sun would be at the same distance. If of the same 

 density, its mass would, therefore, be 8-872 times the solar 

 mass, a result not differing very widely from that found 

 from the orbit. As, however, I have no information of the 

 character of the star's spectrum, I cannot say whether or 

 not it is comparable with the Sun. 



It seems to be stiU very doubtful whether 61 C'ygni is 

 really a binary star, but assuming a parallax of 0-45", and 

 the star's magnitude at 5-11, as measured at Harvard, I 

 find that the Sun is about 8-39 times as bright as 61 Cygni, 

 and its mass, therefore, considerably greater. The star 

 has, according to Professor Pickering, a peculiar spectrum of 

 the solar type. 



Let us now consider the close binary stars recently 

 discovered with the spectroscope, and which are known as 

 -• spectroscopic binaries.'' With reference to Algol, which 

 may be considered as a binary pair, in which one of the 

 components is a dark body. Professor ^'ogel finds that the 

 combined mass of the system is about two-thirds of the 

 Sun's mass. From the dimensions he gives for the com- 

 ponents, I find that their mean density is about one-third 

 that of water, so that they are probably gaseous bodies. 

 As the paraUax of this star has not yet been determined, 

 we cannot say what the Sun's magnitude would be if 

 placed at the star's distance, but as the spectrum of Algol 

 is of the first or Sirian type, we may conclude that it is 

 bright in proportion to its mass. 



For ^ Urs« Majoris (Mizar) Professor Pickering finds a 

 mass eqiTal to forty times the mass of the Sun. KlinixerfiTes 

 foimd a parallax of about U-Ol-")'' for this star. At this 

 dist-ance the Suu would be reduced to a star of only 7*8 

 magnitude. The Harvard measure of t, Uraae is 2-38. It 

 is therefore 5-42 magnitudes, or 147 times brighter than 

 the Sun would be at the same distance. It should there- 

 fore be, if of the same densit}-, 1787 times the mass of the 

 Sun. But the spectrum is of the first type, and the star is 

 therefore not comparable with the Sun in its physical 

 constitution. We have here another example of great 

 brightness in proportion to mass. 



P Aurigfe was discovered to be a close binary with the 

 spectroscope at Harvard Observatory, and the discovery 

 has been fully confirmed by the observations of Professor 

 Vogel at Potsdam. The period is about four days, and the 

 distance between the components about 16 millions of 

 miles. From these data I find that the mass of the 

 system is about five times the mass of the Sim. Eeceut 

 photographic measurements by Professor Pritchard at 

 Oxford have yielded a parallax of 0-059" and 0-065" 



(ohserrntori/, .June, 1891). Taking a mean of these results, 

 or 0-002", we have the Sun reduced to 7*17 magnitude if 

 placed at the distance of the star. /? Aurigfe was measured 

 1-94 magnitude at Oxford and 2-07 at Harvard. We 

 may therefore assume its magnitude at 2-00. This 

 gives a dift'erenee of 5-17 magnitudes between the light of 

 the Sun and that of /i Aurigie. In other words, /S Auriga- 

 is 117 times brighter than the Sun would be if placed in 

 the same jjosition. If therefore of the same intrinsic 

 brightness of surface its diameter would be 10-8 times the 

 diameter of the Sun, and its volume 1265 times the Sun's 

 volume. We see, therefore, that — like Sirius — this star is 

 very much brighter than the Sun in proportion to its mass. 

 As the spectrum of 3 Aurigse is of the first or Sirian 

 type, we have here another example of great brilliancy in 

 proportion to mass, a feature which seems characteristic of 

 all stars of the Sirian type. 



Spica. — The spectroscopic observations of this bright 

 star indicate a mass of about two and a half times the 

 mass of the Sun. The parallax has not yet been well 

 determined (Briosehi found a negative parallax), but 

 judging from its small proper motion, the star's distance is 

 probably very great. As it is a standard star of first 

 magnitude, its brightness would seem to be enormous in 

 proportion to its mass, and here again we have a spectrum 

 of the Sirian tj'pe. 



We may therefore conclude that binary stars with 

 spectra of the first type — and probably all stars of this 

 type — are very bright in proportion to their mass, while 

 those showing spectra of the second or solar tjrpe are 

 intrinsically much less luminous and have a brightness 

 approximately propqrtioual to their mass. 



[Many of the parallaxes made use of by Mr. Gore in 

 these calculations are no doubt extremely doubtful. But 

 in such an enqufr}', even the roughest estimates ai-e of 

 value. The evidence collected tends to indicate that stars 

 of the Sirian type are either less dense than the Sun — that 

 is, that they are in an earher stage of condensation — or 

 that their photospheres are more brilliant, area for area, 

 than the solar photosphere. — A. C. Rany.vrd.] 



Hcttcrs. 



[The Editor does not hold himself responsible for the opinions or 

 statements of correspondents.] 



PERMUTATIONS AXD COlIBrXATIOXS. 

 To the F.ditor of Knowledge. 

 Dear Sir, — It has occurred to me that the solution of 

 interesting questions concerning the number of possible 

 changes or permutations in the arrangement of things is 

 often rendered impossible to most persons in consequence 

 of the great labour involved in the mere arithmetical pro- 

 cess of computation. I am quite aware that the higher 

 mathematics has furnished us with a formula, including 

 functions of tx and c, by means of which the factorials of 

 high numbers are obtainable -n-ith as great a degree of 

 exactness as the use of logarithms will afford ; but, when 

 very great numbers are under consideration we never want 

 exactness — nor could we obtain it if we did. Some 

 numbers are so bewilderingly vast that it would tak.- 

 years even to write them down in figures ; such numbers 

 can only be apprehended by means of their logarithms, 

 or, which comes to the same thing, by the statement how 

 many figures they take. The same thing apphes, though 

 in a less degree, to numbers not quite so enormous. When 



