126 ASTRONOMY. 
gate the direction and velocity with which the body will move forward. 
This is easily done by representing the given directions and velocities by 
the two straight lines, AB, CD (pl. 6, fig. 11), completing the parallelogram 
ABCD, and drawing its diagonal AD. AD, by its position with respect to 
AB and AC, will then give the desired direction, and by its length the 
desired velocity of the motion of the body. This construction, so impor- 
tant in mechanics, is called the parallelogram of forces; AB and AC, the 
lateral forces ; and AD the mean force, the resulting force, or simply the 
resultant. It has been previously mentioned (section 32) that in astronomy 
a very important application is made of the parallelogram of forces. 
Refraction ; Morning and Evening Twilight. 
47. Refraction or bending of rays is of great importance in astronomical 
observations, as it causes the apparent to differ from the true altitude of a 
star. The atmosphere, like any other transparent body, turns an obliquely 
incident ray of light, SA (pl. 6, fig. 17), from its rectilineal direction—in 
other words, it bends it. Thus a ray of light, SA, coming from a rarer 
medium (the ether), and incident at a point, A, upon a denser medium (the 
atmosphere), is bent towards the perpendicular BAC at the point of inei- 
dence, just in proportion to the density of this medium into which the light 
passes. Suppose now an observer to be situated at any point, A, of the 
earth’s surface, KAk (fig. 16); furthermore, let L/, Mm, Nn, represent 
successive strata of decreasing density, into which the atmosphere may be 
supposed to be divided, these evidently being spherical layers congentric 
with the earth’s surface, KAk. Finally, let S represent a star beyond the 
external limits of the atmosphere. If now there were no atmosphere, the 
observer at A would see the star in the direction of the straight line AS. 
In reality, however, the ray SA begins, as soon as it reaches the atmosphere 
at d, to take a more inclined direction, dc, according to the above-mentioned 
law of dioptrics. This change of direction at first, owing to the extraordinary 
rarity of the outermost layers of the atmosphere, is very slight, but increases 
as the ray approaches the earth, entering successively into denser and denser 
strata, and the refraction becoming accordingly greater and greater. Thus, 
instead of following the rectilineal direction SdA, it describes a curve Sdcba, 
which becomes more and more concave, finally reaching the earth, not at A, 
but at a point a, nearer toS. This ray consequently does not come to the 
eye of the observer. The ray by which the observer at A perceives the star 
S, is not SdA, but another ray, which, in the absence of an atmosphere, 
would have reached the point K, behind the observer. Now, however, by the 
refractive power of the atmosphere, it is bent into the curved line SDCBA, 
actually reaching the earth at A. It is a well-known law in optics, that every 
object is seen in the direction which the ay from the object has at the time 
of its entrance into the eye, the intermediate course of the ray not coming 
into account. The star 8 will therefore be seen, not in the direction AS, 
but in the direction of the straight line As, tangent to the curve SDCBA 
126 
