June .o, 1890.] 



SCIENCE. 



375 



at the bottom of Ibe Rbone valley, and will to morrow be sweep- 

 ing over the Rhine, or vice versa, according to the direction in 

 which the surface wind blows. Hence Dr. Hann's observations, 

 however valuable otherwise, can have only small bearing on the 

 question of the cause of cyclones and anticyclones. 



Franz A. Velschow, C.E. 



Jones Point, N.Y., June 2. 



On the Determination of Parallax by the Spectroscope. 



In the winter of 1883-84 it occurred to the writer that the 

 spectroscope might be made use of in the determination of the 

 parallaxes of certain double stars. As there were no data at hand 

 that would allow a numerical example to be worked out, the 

 method was not published at the time, but was withheld until 

 such data should be available. Recently my attention was drawn 

 to the systematic measures carried on at Greenwich since 1876; 

 and, although those are very unsatisfactory on account of their 

 large probable error, it may be of interest to apply them to an 

 actual parallax determination. 



The method about to be proposed is based upon the well known 

 fact that the positions of the lines in a star's spectrum depend not 

 only upon the substances to which these lines are due, but also 

 upon the velocity of Ihe star's motion in the direction of the line 

 of sight. So far as the writer can see, it is applicable only to 

 double stars; and it may be made use of in two different forms, 

 the first of which is applicable when both components of the star 

 are bright enough to be observed spectroscopically, and the sec- 

 ond when only one component is bright enough to be so observed. 



In the flr?t case, both components being bright, let S be the one 

 to which the orbit is referred, and let C be the companion; u is 

 the angle that the tangent at C makes with C S, and 6 the angle 

 that it makes with the line of sight. V„ is the velocity with which 

 S is receding from the earth at a given moment, and F, is the 

 velocity with which C is receding at the same moment, both 

 being expressed in miles per second. The orbital velocity of G at 

 this riioment we will call v, the unit of length being that length 

 which subtends an angle of one second at the star's distance from 

 the earth. If tt is the parallax of the star (supposed unknown), 

 and Z)is the radius of the eai-th's orbit, v can be expressed in miles 



per second by multiplying it by — . Expressing it in this man- 

 ner, we have ^ 



(Fo — V,) = V . — . cose (1) 



But 



2 A 



(2) 



p r sm a 



where p is the period of the star in seconds of time, r is the radius 

 vector of the component in seconds of arc, and A is the area of 

 the oi-bit, the unit of length being the same as in the case of v. 

 Substitution in (1) gives us 



2 A D cose 1 



(^0 - T',) = : 



p r sm I 



(3) 



The first member of this equation is to be observed by the spec- 

 troscope, and the co-etHcient of the second member is to be com- 

 puted from the elements of the star's orbit The only quantity 

 remaining is the parallax of the star, which is found by simple 

 division. If it is desired to make a number of observations at 

 different times, and combine the whole by the method of least 

 squares, the normal equation will be, of course, [a^]a;=[ay, equa- 

 tion (ij) being now of the form ax=l. 



Undoubtedly the best way to determine the absolute term in 

 equation (3) is to photograph the spectra of both star's on the same 

 plate, and measure ihe intervals between the corresponding lines 

 in the two The probable error of a determination so made will 

 be less than if Vo and Fj were measured separately and their 

 difference taken. I do not find that this has been done in the 

 case of any star whose orbit is known; but that the lines in the 

 spectrum of a double star can be so photographed and measured, 

 at least in certain cases, is well shown by Professor Pickermg's 

 recent work on Beta Avrigce and Zeta Vrsce Majoris, which stars 

 were not known to be double until the spectroscope showed them 

 to be so. It is true that the proximity oPthe components of these 



stars, and their consequent short periods, make the measurement 

 particularly easy in these cases ; yet I trust that it is not unrea- 

 sonable to hope that measures may be made on other stars suffi- 

 ciently good to afford us some idea of their parallax. 



When one of the components of a double star is so faint that its 

 spectrum cannot be observed, it becomes necessary to modify the 

 foregoing mode of procedure somewhat. Let S be the principal 

 star, as before, and C the companion. Let Fbe the velocity of 

 recession of the principal star, and Vo the velocity of recession of 

 the centre of gravity of the system (F„ being appreciably constant 

 for many centuries). Let a be the semi axis major of the orbit of 

 the companion, when referred to the principal star, and let a, be 

 the semi-axis major of the smaller ellipse described on the heavens 

 by the larger star, in consequence of its having a companion (this 



may be determined by comparing the position of the principal 

 star with smaller stars in the vicinity, not physically connected 

 with it) . Then we have the equation 

 tti D 



n + 



V . cos e 



(4) 



(this is obtained by resolving the velocity of the component along 

 the line of sight, multiplying the result by a^-^a to find the cor- 

 responding differential velocity of the principal star, and adding 

 to the velocity of the centre of gravity of the system). Substitut- 

 ing in (4) the value of v as givein in (2), we have 



„ „ , a, 2ADcose 1 ,_, 



F = Fo -I- J- . : . — (5) 



a p r sm u 77 



This is the form of the observation equation. Fis observed, at 

 intervals, by the spectroscope, and corresponding values of the 



co-efficient of — are computed. The normal equations are, then. 



[pV] 



[PFV]: 



[p]V, +[pF]x, 

 [Pn Vo +IPF''] X, 



(5) being of the form F = V^ + F x. 



As already intimated, the writer has applied this method to a 

 particular case, using the spectrum observations made at Green- 

 wich since 1876, together with one measure obtained by Huggins 

 in 1868. Sirius was selected for the purpose for several reasons. 

 Its orbit is fairly well known, the spectrum observations on it 

 cover an interval of twenty years, the period of the star is short. 

 and various determinations of its parallax have already been 

 made by the direct method. The elements of the star, according 

 to Mr. J. E. Gore (Monthly Notices of the Royal Astronomicat 

 Society for June, 1889), are as follows: — 



T = 1896.47 a = 49° 59' (1880.0) 



P = 58.47 i = 5.50 23' 



a = 8".58 7 = 210° 18' 



e = 0.4055 ^ = -6". 157 



It appears also, from Auwers's work, that the semi-axis major of 

 the orbit that the principal star describes about the centre cf 



