1907-8.] Sunset and Twilight Curves, and Related Phenomena. 313 
Putting, therefore, \ = x, we have the two equations 
cos y=- tan tan S , 
sin S = sin e sin x , 
which, when S is eliminated, give the equation of a curve of approximate 
sunset (apparent time) referred to rectangular axes.* 
A rougher approximation, which leads to simpler results, can be 
obtained by supposing N to move uniformly. Then, putting TN = ®, 
we get 
tan 8 = tan e sin x , 
and the equation of the curve assumes the simple form 
cos y = — tan <£ tan e sin x . 
§ 4. It will be sufficient, in the first place, to take the latter approxima- 
tion. We have then the equation 
cos y = k sin x . 
This is the equation of a repeating curve, and the portion contained 
Q 
between x = — ^ and and y = 0 and ir represents approximately the graph 
A A 
of the time of sunset for one year from one winter solstice to another. 
The form of the curve depends upon the value of k. k = ±1 divides 
the family of curves into two classes. 
For k = ±1 
cos y= ± sin x = cos 
hence _ + x = 2mr ± y , 
and we have two systems of straight lines 
x ± y — mr + ^ . 
Let | k | < 1. 
When x = mr , y = mir + ^ , 
A 
x = 2ft7r + „ , y = 2mTr ± cos 1 k, 
A 
* The effect of the equation of time, reducing apparent time to mean time , will be most 
marked in low latitudes, and it will produce a skewness in the curve, displacing the 
maximum towards the autumnal equinox and, if the latitude is sufficiently small, 
producing another maximum a little before the vernal equinox. The minimum will be 
intensified and displaced also towards the autumnal equinox. In what follows we shall 
•consider only apparent time. 
