5^6 LECTUUK XLIV. 



sun's path at the time of the equinoxes, or the apparent place of the equinoctial 

 in the heavens, for any given point on the earth's surface, we may represent the 

 sun's path at any other time by a smaller circle parallel to it. Speaking 

 however, more correctly, the sun's apparent path is a spiral, formed by the 

 continuation of these supposed circles into each other. 



The effect of the centrifugal force, derived from the earth's rotation, is 

 perceptible, at the equator, in the retardation of the vibrations of pendu- 

 lums. The whole centrifugal force at the equator is found, by computation, 

 to be -i^T of the force of gravity ; tut the diminution of the force of gravi- 

 tation appears, by experiments on pendulums, to be T-'-g-; this diminution 

 being the sum of the centrifugal force, and of the decrease of gravity on 

 account of the oblate figure of the earth, the equatorial parts being further 

 removed from its centre, and the force of gravity being less powerful there. 

 The changes of inclination in the earth's axis are observable in the places of 

 the equinoctial points, and in the situation of the plane of the earth's equator 

 with respect to the fixed stars; and the secular diminution of the obliquity 

 of the ecliptic is discoverable by a comparison of distant observations 

 on the sun's apparent motion, and on the places of the fixed stars with 

 respect to the ecliptic. 



For the phenomena of twilight, we are principally indebted to the light 

 reflected by the atmosphere: when the sun is at a certain distance only below 

 the horizon, he shines on some part of the air immediately visible to us, 

 which affords us a portion of reflected light. The distance, at which this 

 may happen, has been variously estimated, and it is perhaps actually dif- 

 ferent in different climates, being a little greater in countries near the poles 

 than in those which are nearer the equator: there is also sometimes a second- 

 ary twilight, when the parts of the atmosphere, which reflect a faint light on 

 the earth, are themselves indebted for this light to an earlier reflection. 

 Some have assigned 18° as the limit of twilight,and on this supposition, allow- 

 ing for refraction, the atmosphere must be capable of reflecting sensible 

 light at the height of about 40 miles. Mr. Lambert, on the contrary, makes 

 the limit only about 6^°. The duration of twilight is greater or less as the 

 sun moves more or less obliquely with respect to the horizon ; it is, therefore, 

 shortest near the time of the equinoxes, since the equinoctial intersects the 



