ON THE RETARDATION OF SUNRISE*. 



SUPPOSE the sun on the eastern horizon : his actual apparent 

 motion until next sunrise may obviously be replaced by a ficti- 

 tious one composed of an unscrew rotation round II, the pole of 



z 



the ecliptic, combined with a screw rotation round P, the pole of 

 the earth, the magnitude of the latter rotation measuring the 

 retardation. Combine these rotations supposed small into one : 

 the resultant axis must lie in some point Q, where PIT produced 

 cuts ZS, (Z the zenith, S the sun). The angle between the 

 component rotations is constant, being PEE or w. Therefore 

 the ratio of inequality between the given rotation round II, 

 measured by the sun's diurnal motion in his orbit, and rotation, 

 round P, which measures the retardation, is then greatest when 

 the resultant axis Q lies as near as possible to II, or, in other 

 words, when PQ is a minimum. Now, S being the pole of the 

 arc ZTl, it will easily be seen that the angle $^11 is a right 

 angle and the angle QZP is an obtuse angle, while near the 

 equinox the angle ZQP must always be acute. Therefore PQ 

 is least when Q lies in PZ, and then II does so too. In other 

 words, the retardation is least when II culminates at sunrise, 

 that is, at the equinoxes. 



The common expression for retardation may easily be de- 

 duced from the expression furnished by what has been said, 

 sin II Q 



viz. 



sin PQ 



Now first published. 



