3. _Changes in the position of a star._ From the positions of a star on two or more occasions we obtain its apparent motion, also called the _proper motion_ of the star. We may distinguish between a _secular_ part of this motion and a _periodical_ part. In both cases the motion may be either a reflex of the motion of the observer, and is then called _parallactic_ motion, or it may be caused by a real motion of the star. From the parallactic motion of the star it is possible to deduce its distance from the sun, or its parallax. The periodic parallactic proper motion is caused by the motion of the earth around the sun, and gives the _annual parallax_ (π). In order to obtain available annual parallaxes of a star it is usually necessary for the star to be nearer to us than 5 siriometers, corresponding to a parallax greater than 0″.04. More seldom we may in this manner obtain trustworthy values for a distance amounting to 10 siriometers (π = 0″.02), or even still greater values. For such large distances the _secular_ parallax, which is caused by the progressive motion of the sun in space, may give better results, especially if the mean distance of a group of stars is simultaneously determined. Such a value of the secular parallax is also called, by KAPTEYN, the _systematic_ parallax of the stars. When we speak of the proper motion of a star, without further specification, we mean always the secular proper motion. 4. Terrestrial distances are now, at least in scientific researches, universally expressed in kilometres. A kilometre is, however, an inappropriate unit for celestial distances. When dealing with distances in our planetary system, the astronomers, since the time of NEWTON, have always used the mean distance of the earth from the sun as universal unit of distance. Regarding the distances in the stellar system the astronomers have had a varying practice. German astronomers, SEELIGER and others, have long used a stellar unit of distance corresponding to an annual parallax of 0″.2, which has been called a “Siriusweite”. To this name it may be justly objected that it has no international use, a great desideratum in science. Against the theoretical definition of this unit it may also be said that a distance is suitably to be defined through another distance and not through an angle--an angle which corresponds moreover, in this case, to the _harmonic_ mean distance of the star and not to its arithmetic mean distance. The same objection may be made to the unit “parsec.” proposed in 1912 by TURNER. For my part I have, since 1911, proposed a stellar unit which, both in name and definition, nearly coincides with the proposition of SEELIGER, and which will be exclusively used in these lectures. A _siriometer_ is put equal to 10^6 times the planetary unit of distance, corresponding to a parallax of 0″.206265 (in practice sufficiently exactly 0″.2).