154

THE MOON’S CHANGES

the mean interval being only about 50| minutes.And the full moon is near the ascending node of the

books of astronomy. But it should be noticed that Ferguson didnot compute the values in this table, hut only estimated the values“ as near as could be done from a common globe, on which themoon’s orbit was delineated with a black-lead pencil,” and he wasnot successful even in his application of this very rough method,by which, or by a simple method of projection, it may readily beshown that the maximum difference is greater and the minimumdifference less than Ferguson supposed. If the eccentricity of themoon’s orbit and her consequently variable motion be taken intoaccount, a yet greater difference results. It is easy to obtain equa-tions whence we can calculate the difference in the hour of risingunder the circumstances in question. They are as follows, theassumption being made that the moon is crossing the equator atrising :—Let a be the inclination of the moon’s path to the equator(a ranging in value between 28° 44' and 18° 10'), l the latitude ofthe station. Then let h be the moon’s mean hourly motion on theecliptic (about 3&| minutes of arc), x the time in hours betweenher rising on the day when she is on the equator and on the nextday. Then her motion on the ecliptic is x h. Put =Take then

sin tj/ = sin a sin 9 (i)

and sin <j> = tail l tan (ii)

Then ip is approximately the hour-angle by which the intervalbetween successive risings exceeds or falls short of the meaninterval (1 d. 50^ m.). So that

x = 24'84 ± ^; that is 9 = 24'84 h + 0

= 373° ± ^ approximately.

These equations are theoretically sufficient to determine 6 (or z) i

but practically, it is sufficient to adopt a value of j- (half an hour is

fl

near enough), giving x = 24'34 ; 0 = 12° 22J' about. Then use (i)and (ii) to calculate <p, and repeat the process, using in it the valueof <j> thus deduced.