TRANSACTIONS OF THE SECTIONS. 35 



enough to allow of accurate observation ; whereas the method of determining the 

 distance of a star by its parallax becomes more difficult as the distance of the star 

 increases, notwithstanding any brightness which it may have. 



The method now proposed is founded on that of spectral analysis. I suppose a 

 certain ray, which I will call X, to be chosen as the standard ray, and to be care- 

 fully observed at various times in each of the stars of a binary system during an 

 interval of some years. The orbit described by the stars around their common 

 centre of gravity must not lie in a plane perpendicular to the visual ray joining 

 those stars aud the earth, nor must it approach that position too nearly, otherwise 

 the true result would be masked by the errors of observation. The simplest case 

 is that of two stars, equal in mass and brightness, and revolving in circles about 

 their common centre of gi-avity. Supposing such a system of two stars to exist, 

 the most favourable case is when the plane of their motion passes through the 

 earth. If it does so, the stars will appear to move in straight lines. Supposing 

 them to be, when fii-st observed, at their greatest elongation, they will approach 

 each other with an increasing apparent velocity, varying as the sine of the time (or 

 circular arc described) until they come into apparent conjunction, when one star 

 will be hidden by the other for a certain time, after which they will recede from 

 each other in like manner as they had approached. But the observer would not 

 be able to say with certainty which of the two stars was nearest to him, since the 

 same phenomena would be presented if the distances of the two stars were inter- 

 changed, and at the same time the direction of their motions reversed. Now 

 suppose the method to be applied which I have proposed. At the time of their 

 conjunction, or near it, neither star would be approaching the earth, consequently 

 the observed deviation of the ray X (if any) from its normal position would be due 

 to the 2)ro2)er motion of the system of the two stars relatively to the earth, which is 

 a constant quantity to be allowed for in all other observations. Now suppose another 

 set of observations to be made at the time of the greatest elongation of the two 

 stars. At that time each of the stars is apparently stationary, but in fact one 

 of them is approaching and the other receding from the earth with a maximum 

 velocity. The observed deviation of the ray X will therefore be different in the 

 spectra of the two stars, and (allowance having been made for the proper motion of 

 the system) it wiU appear at once which of the two stars is approaching the earth, 

 and the question of its direct or retrograde orbit will be resolved. At the same time 

 the distance of the two stars from the earth will result from the calculation. It 

 will be well, perhaps, to take a hypothetical example, which will show how this 

 element results from observation. 



I suppose, then, that observation has shown : 



(1) The period of one complete revolution of the binaiy star round its centre of 

 gravity to hejiftt/ years. 



(2) The greatest elongation of the stars to be ten seconds. 



(3) And at the time of this gi-eatest elongation the deviation of the ray X to be 

 such as to prove that one of the stars is then approaching the earth at the rate of 

 ten miles per second, and the other star receding from the earth at the same rate. 

 And this will evidently be their true velocity in their orbit. 



Now 50 years = 1,577,880,000 seconds, and therefore since each star moves in 

 its orbit at the rate of ten miles per second, it describes in the course of one whole 

 revolution of 50 years a circle of 15,778,800,000 miles in circumference. The 

 radius of this circle is the distance of the star from the common centre of gi-avity, 

 and therefore the diameter of the circle is the distance of the two stars from each 

 other (which in the hypothetical example I have selected is constant). This 

 diameter will be found to be about equal to 54 radii of the earth's orbit. Now, 

 when the stars were at their greatest elongation, observation showed their angular 

 distance to be teji seconds. Consequently we have only to calculate at what distance 

 from the earth a length of 54 radii woidd subtend an angle of 10", and we find 

 that this would occur at a distance of 1,113,500 radii. Such, then, is the distance 

 of the binary star from the earth, namely, 1,113,500 times the distance which sepa- 

 rates the earth from the sun. 



So_ simple a case as the hypothetical one which I have here calculated is, indeed, 

 not likely to occur in practice ; most cases would require a gi-eater complexity of 



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