December 1, 1892.] 



KNOWLEDGE 



225 



along HB: In Fig. 2, however, the corresponding effect 

 would be much less marked. To these considerations may 



be added that pointed out by the late Mr. Proctor, that 

 the galactic coal-sacks and lacunae are altogether against 

 any great depth along the line of sight. 



-A 



It seems to me that the strength of this argument (such 

 as it is) gives us something more than a clue to where the 

 greatest proportion of the mass of the Milky Way may be 

 found. 



Let us, at a venture, in the absence of any trustworthy 

 evidence one way or the other, make a similar assumption 

 to Sir William Herschel's, and assume the stars to be 

 uniformly distributed throughout the whole Milky Way. 

 Let P, Q (Fig. 3) be lines of sight bounding any 



p 



galactic area. The actual distance of this area is imknown, 

 it may be at A or at B, or since the average angular ; 

 breadth of the galaxy is nearly constant everywhere, let , 

 us assume that A is a portion of it from one part of the ! 

 sky at a certain distance, and that B is a portion from i 

 another part of the sky at another distance. A and B are 

 of the same angular dimensions. We want to compare 

 the brightness of A and B on the assumption that each 

 contains the same number of stars in a given volume. I 

 Let d and D be the true distances of A and B, B and A 

 their diameters ; then 



Volume of A : volume of B = 8^ ; A' 



= d' : D'' 

 = d^ : n'd' 

 = 1 -.n" 

 where D = n d. 



But the brightness of each star in B will be reduced in 

 the proportion of D - : d-, that is as n-d- : d-, that is as 

 n - : 1. Therefore the brilliancy of B will be to that ot A 

 in the proportion of n : 1. In other words, on the 

 assumption of uniformity of distribution, the brightness of 

 the different parts of the Milky AVay would vary as the 

 distance. It has been shown (see Knowledge for February, 

 1891) that the brilliancy of a given sheet or volume of 

 stars will remain constant whatever its distance ; and this 

 fact has been used to controvert Proctor's spiral theory of 



* Nor would the optical effect be materially diifcreut on the view 

 that the Milky Way is a collectiou of clouds, oue behind the other, 

 for a vast distance along the line of sight. I do not contend that it 

 has not this particular cloud structure ; what seems certain is, that if 

 it has, the clo\ids can neither be far apart nor numerous, uor of 

 great individual depth. Bat see Mr. Banyard's article in KauVfLBUQE 

 for July, 1890.— J. R. Sutton. 



the Milky Way. For the main argument for the spiral 

 theory was based upon the erroneous conclusion that the 

 faintness of the great branching streams fi-om the Milky 

 Way was due to an approach to " evanescence through 

 vastness of distance." But on the assumption of a 

 uniformity of distribution, it would be necessary (since the 

 angular breadth of the great branching streams is not 

 materially less than that of the main stream) that these 

 streams, if part of a spiral, should be considerably 

 brighter than the rest of the galaxy, f unless, indeed, the 

 distribution decreased with distance from the sun : an 

 arrangement not altogether impossible perhaps, if we 

 could regard the sun as in some way the mainspring of the 

 whole visible universe. 



It will be noticed, however, that the calculation above 

 involves, in fact. Sir William Herschel's other and later 

 hypothesis of a general equality of real size among the 

 stars. If the Milky Way stars are collected into groups of 

 difl'erent orders, i.e., if stars of one size segregate into one 

 region, those of another size into another region — a ten- 

 dency which seems to obtain, to some degree, among the 

 sporadic stars in various parts of the sky — a considerable 

 modification must be introduced into our result. Con- 

 sider two stars S and s, of masses in and 1, and diameters 

 A and S respectively, at equal distances from the sun. 

 Let their brilliancy, surface for surface, be the same. 

 Then we have — 



The total brightness of S ; the total brightness of s as the 

 surface of S : surface of s, that is as A- : 8-, but m : 1 = 

 A ^ : 8 "> if the densities are the same. 



Therefore A- : 8- = mf : 1. 



Therefore the total brightness of S : the total brightness 

 of s as m| : 1 . 



If now we could suppose the iikius of any given volume 

 of the Milky Way to be constant, for every star (S) in the 

 one place we should have m stars (s) in the other place. 

 Hence — 



The total brightness of S : the total brightness of m s 



= mf : m 

 = 1 : mj 

 so that, on the assumption of equal surface brilliancy, the 

 region containing the smaller stars would be brighter in 

 the ratio mj to unity, and it follows that the ratio of the 

 brightness of A to the brightness of B (referring again to 

 the notation of Fig. 3) will be as unity is to n m^, or as 

 )»J ; H, according as the smaller stars are found in B or A. 

 In the latter case, if m be greater than n^, A would be 

 brighter than B. If, then, the galactic stars collected 

 rigorously in classes of magnitude, it would be clear that 

 the brightness, taken alone, of the Milky Way, would have 

 no immediate significance in the present state of our 

 knowledge. 



That the galactic stars do collect into minor sheets and 

 volumes of one particular star-magnitude, may be admitted 

 without prejudicing the fact that over any extended region 

 such a tendency is not obvious. Individuahty, so to speak, 

 becomes lost in the crowd. Photography adds its testi- 

 mony to telescopic observation, that the average magnitude 

 of the stars belonging to the Milky Way proper, in any 

 given region, will be about equal to the average star- 

 magnitude of the whole ring. That is to say, if we take a 



[t This ingenious reasoning of Mr. Sutton leaves out of account 

 the absorption of light in its transmission to us from distant parts of 

 space. It seems very improbable that the loss of light due to absorp- 

 tion shoidd just balance the increase of brightness of the stream duo 

 to its greater thickness, especially when great distances are involved, 

 for the absorption would increase accordinj to an exponential law . 

 while the brightness would, accoi'din^ to Mr. Sutton's assimiptions, 

 oidy increase directly iis the distance. — A. C Ranyaed.] 



