338 Scientific Proceedings, Royal Dublin Society. 



appearance of twilight, and then perhaps handed down from 

 one person to another, we could scarcely expect any great 

 accuracy. 



It will be observed that the greatest discrepancies between the 

 clock curves and those I have computed occur in the summer half 

 of the year. This arises from the fact that the clock curves are 

 symmetrical with regard to the equinoctial line, whereas they ought 

 to rise more abruptly in the summer than in the winter half of the 

 year. This want of symmetry is due to the fact that we are not 

 dealing with the time of the sun's appearance on a great circle such 

 as the horizon, any changes in which at one side of the equinox we 

 should expect, neglecting the eccentricity of the earth's orbit, to 

 find counterbalanced by corresponding changes at the other side ; 

 but we have to do with the time of its appearance on a small circle 

 13° below the horizon, the changes in which will not take place 

 symmetrically in the two periods. This will be seen more clearly 

 from the following formulae. If we want to determine the hour- 

 angle of the sun when on the horizon, we have, if § is its declina- 

 tion, the latitude, and h the hour-angle, 



cos h = - tan <£ tan S. 



Now to change the sign of § in this merely changes the sign of 

 cos h, so that for each value of h, corresponding to a particular value 

 of 8, we have 180 - h corresponding to - B. 



On the other hand, if we want to determine the hour-angle of 

 the sun when at a distance x below the horizon, we have 



, - sin <f> sin S - sin x ,, , 



cos h = - s ; (1) 



cos <p cos 6. 



in which, by changing the sign of S, we may or may not alter the 

 sign of cos h, but we also alter its absolute value. 



It appears to me very likely that a primitive people, having 

 observed the changes in the length of the day throughout the 

 year, would assume that any want of symmetry in the curve they 

 obtained, if they noticed it at all, was due to errors of obser- 

 vation, and would accordingly take a series of mean values to suit 

 both halves of the year from which to construct their curves. 



