666 Proceeding f^ of the Boyal Irish Academi/. 



in which (^denotes the angle between the tangent and the major-axis; 

 X , , , , , , the line of nodes , , , , 



y ,, ,, ,, the plane of orbit and tangent 



plane to sphere ; 

 P ,, period in years, \ 



n „ parallax in sees, of arc, ( ^^ ^^^ ^^^^ . 



V ,, the velocity in miles per se- ( 



cond in the line of sight, / 



and I ,, mean motion of the Earth in miles. 



The equation (5) gives us, then, a relation between IT and V, de- 

 pending only on the period and the angular elements of the orbit. If 

 one is measured, we can at once determine the other. If the quan- 

 tity k be greater than unity at any time, we know one of two things — 

 either that the parallax is not less than one-tenth of a second of arc, 

 or that the velocity in the line of sight is not less than ten miles a 

 second. If, then, on turning the spectroscope on the star we find that 

 the lines in both spectra are absolutely coincident in position, we know 

 that the parallax is not less than one-tenth of a second, and will repay 

 all the care that may be bestowed on its determination. On the other 

 hand, if there is any displacement visible, the measurement of its 

 amount will give us the value of V, and consequently of 11. Thus all 

 double stars, for which k is at any time greater than unity, may be 

 said to be within measurable distance either by the spectroscopic or 

 the trigonometrical method. 



On the other hand, if k is less than unity it does not follow tliat 

 the distance defies measurement, for a sensible value of 11 may be 

 counterbalanced by the smallness of the linear dimensions of the orbit, 

 or the length of the period, giving a small value for V; whereas if k 

 is less than unity, and Fis found of any considerable magnitude, it 

 informs us with a certainty, which the mere failure to measure its pa- 

 rallax trigonometrically could never reach, that the star is at an 

 inconceivable distance from the solar system. 



All this is on the supposition that a parallax not greater than 0""1 

 could be measured by the best modern methods, and that the spectro- 

 scope could detect a velocity as small as ten miles per second. 



I have, purposely, taken a veiy small limit for the parallax — for 

 this reason, that if it is ascertained by other means that IT is not less 

 than one-tenth of a second, an observer may perhaps take more extra- 

 ordinary precautions, and may extend his obseiwations over a longer 

 period than if the nature of the ultimate result was doubtful, since he 

 knows he must, if his observations are correct, get a sensible value for 

 the parallax. jS^or is ten miles pur second, I think, too small a limit 



