170 



NATURE 



{June 22, 1882 



point for every observed angle of position, the epoch in 

 years and decimals being measured as an abscissa along 

 the horizontal lines, and the angle in degrees as an ordi- 

 nate along the vertical ones. A series of points are thus 

 obtained, which, if the observations were exact, would 

 necessarily admit of a regular curve being drawn through 

 them, whose nature is of course determined by the laws 

 of elliptic motion, and one of whose essential characters 

 is to have within those limits of the abscissa, which 

 correspond to a whole period of revolution (that is, to a 

 difference of 360 units in the ordinates), in some cases 

 two, in some four, points of contrary flexure, but never 

 more than the latter, nor fewer than the former, and to 

 have, moreover, in all its points, a peculiarly graceful and 

 flowing outline. The errors of observation, however, 

 prevent the drawing of such a curve through all the 

 points. It must be drawn with a free but careful hand, 

 not through, but among the points, and so that it shall 

 deviate less from every point, according as it is more or 

 less reliable. Now after Herschel's time the accuracy of 

 the observed distances has wonderfully improved, and we 

 are therefore able to draw another curve representing the 

 distances as ordinates, which then ought to agree with 

 those deduced from the angles, and the angles ought to 

 agree with those deducible by aid of integral calculus 

 from the distances. The curves must be varied till they 

 thus mutually support each other, and then we may 

 construct any number of points of the apparent orbit by 

 reading off the angles and distances for the corresponding 

 epochs on the curves, and if we find the arc described 

 sufficiently extensive, the apparent ellipse is simply drawn 

 as nearly as possible through them. From the apparent 

 orbit the elements of the real orbit, described in space, 

 are then determined. These are seven in number : — 



a. The major semi-axis, expressed in seconds of arc, 

 i.e. the angle under which their mean mutual distance 

 would appear if placed perpendicular to the line of vision, 

 ie. the straight line joining us with the star. 



(. The eccentricity of the real ellipse. 



y. The inclination of the plane of the real orbit to the 

 plane perpendicular upon the line of vision. 



A. The node, i.e. the angle of position of the line in 

 which the plane of the orbit intersects the plane perpendi- 

 cular upon the line of vision. 



n. The longitude of the projected peri-astron, i.e. the 

 angle of position of the companion at the epoch of its 

 actual nearest approach to the main star. 



T. The epoch, when the nearest approach in space 

 occurs. 



P. The period of revolution, i.e. the time it takes the 

 companion to complete an entire revolution round the 

 main star. 



It is impossible to say what part of the orbit is inclined 

 towards us, and what is removed from us, 1 we cannot 

 therefore, distinguish between an ascending and a de- 

 scending node. 



Both before and after Herschel's investigations, several 

 methods of calculating double-star orbits have been pro- 

 posed. Savary, at Arago's request, was the first who gave 

 an analytical determination of an orbit from five points. 

 He also proposed a method, subsequently improved by 

 Encke, for calculating the seven elements of the real 

 orbit from four complete observations. These furnish 

 eight co-ordinates corresponding to known epochs, and 

 as only seven quantities, the elements are sought, the 

 problem is over-determined. It has to be solved with 

 different values of one of the data, in order that all the 

 other data may be represented. The case is, at the time 

 it generally was possible to obtain but three complete 

 positions from the discussion of modern observations, 

 while the fourth, depending upon Herschel's measures, 



: any time observe the speed in 

 approaches to or recedes from 

 naktng some progress ' 



1 This could be ascertained if we c 

 liles per second with which the com 

 1 the line of visi. n. Spectroscopic 



lies, but their apparatus 



: for our purp^ 



gives seldom even an approximate value of the distance. 

 After that, a longer series of angles had become available. 

 Klinkerfues proposed to determine at once the seven 

 elements from six observed angles, and at least one dis- 

 tance. The dimensions of the orbit could evidently not 

 be obtained without the aid of observed dimensions, 

 though all the other elements can be derived from the 

 angles. It sometimes occurs that the companion moves 

 in an orbit, the edge of which is turned towards us. The 

 inclination is then about ninety degrees, and all the 

 angles are nearly equal to, or half a circumference dif- 

 ferent from the longitude of the node. We are then 

 obliged to make more extensive use of the distances. 

 Several analytical methods have been adopted to this con- 

 tingency, which has actually occurred in a few cases. It 

 seldom happens that an orbit founded upon a few posi- 

 tions, even if these are distributed over a great part of 

 the arc described, is the most accurate that could possibly 

 be deduced from all the observations at our disposal. It 

 is therefore advisable to finally correct the elements ac- 

 cording to the rules of the calculus of probabilities, so 

 that the errors ultimately left behind in the representation 

 of the observed places, may, taking into account their 

 relative accuracy and their different peculiarities, be as 

 small as possible. In this country, Hind, one of the 

 greatest authorities on anything relating to double stars, 

 has made most extensive calculations of this nature. 



The number of those double stars whose orbits can be 

 computed is limited to between thirty and forty. The 

 time during which micrometric measures have been made 

 is so short, that one but seldom can make anything like a 

 correct guess of what the whole orbit may turn out to be 

 like, when the period is above three hundred years ; and 

 even so, the peri-astron passage must happen to fall 

 within a hundred years from now ; that is, if the eccen- 

 tricity is not unusually snull. The eccentricity of these 

 orbits is generally large, and it appears to be larger the 

 greater the period and the greater the axis is. N ow in 

 the cases — by far the most frequent —where the measures 

 do not embrace a larger part of the orbit, we represent 

 the motion of the companion by formula;, deduced in 

 accordance with the proportionality of the areas with the 

 times. These formulas suffice to foretell the co-ordinates 

 for some few years to come, and they are interesting in 

 other respects, too. Thus, if the distance of a double 

 star from the solar system were known to us, we would 

 also know the dimensions of the orbit in miles, and then 

 we could calculate its mass by aid of Kepler's laws. Now 

 their distances from us are not known, but if we assume 

 the mass to be on an average e.g. about three times the 

 mass of the sun, then we obtain — if we make the further 

 assumption that the distance actually measured in seconds 

 of arc between the components is in an average of a very 

 great number of stars equal to their mean distance from 

 each other — from the period of revolution (concluded from 

 the variation of the angle) through the inverse calculation 

 their distance from us. Now these hypotheses can only 

 be used in discussing the results of an average of a great 

 number of systems that have certain characters in 

 common, e.g. brightness, colour, or distance. I am at 

 present engaged with such researches, from which I hope 

 to arrive at interesting results. 



Most of the double stars, that so far have been found 

 to revolve, are close and more or less difficult objects. 

 Few of them are more than six and a half seconds of arc 

 asunder. The components are in most cases of about the 

 same brightness, but the companion is, on the whole, 

 smaller, the greater the distance. 



There are two kinds of revolving double stars. The 

 first of them consists of bodies whose colours are strictly 

 identical, whereas the second consists of bodies whose 

 colour is generally complementary. The principal star in 

 both cases is white or yellow — white stars preponderating 

 in the first case, yellows in the second. In systems of 



