It appears, then, that the first or primary bow is formed by two refractions and one reflection; but there is frequently a second bow, on the outside of the other, which is considerably fainter. This is produced by drops of rain above the drop we have supposed at A. If B (fig. 35.) represent one of these drops, the ray to be sent to the eye enters the drop near the bottom, and suffers _two refractions_ and _two reflections_, by which means the colours become reversed, that is, the violet is lowest in the _exterior_ bow, and the red is lowest in the _interior_ one, and the other colours are reversed accordingly. The ray T is refracted at R: a part of it is reflected from S to T, and at T it suffers another reflection from T to U. At the points S and T part of the ray _passes through_ the drop on account of its transparency, towards W and X, and therefore we say that _part_ only of the ray is reflected. By these losses and reflections the exterior bow becomes faint and ill-defined in comparison of the interior or primary bow. In this case the upper part of the secondary bow will not be seen when the sun is above 54° 10´ above the horizon; and the lower part of the bow will not be seen when the sun is 60° 58´ above the horizon. [Illustration: _figure 36._] For the further illustrations of this subject, we may introduce the following section of a bow, (fig. 36.) and, in order to prevent confusion in attempting to represent all the different colours--let us suppose only three drops of rain, and three different colours, as shown in the figure. The spectator O being in the centre of the two bows, here represented,--the planes of which must be considered as perpendicular to his view--the drops A,B, and C produce part of the interior bow by two refractions and one reflection as stated above, and the drops D,E,F will produce the exterior bow by two refractions and two reflections, the sun’s rays being represented by 3,3. It is evident that the angle COP is less than the angle BOP, and that the angle AOP is the greatest of the three. The largest angle, then, is formed by the red rays, the middle one consists of the green, and the smallest the purple or violet. All the drops of rain, therefore, that happen to be in a certain position with respect to the spectator’s eye, will reflect the red rays, and form a band or semicircle of red, and so of the other colours from drops in other positions. If the spectator alters his station, he will see a bow, but not the same as before; and if there be many spectators, they will each see a different bow, though it appears to be the same.