Section III, 1888. [ 17 ] Tkans. Roy. Soo. Canada. 



II. — Occultations of Fixed Stars by ike Moon : Prediction for a given place by a 

 Graphical Method. By W. F. King, B.A. 



(Communicated by E. Deville, May 23, 1888). 



The distance of the star being so great as to be practically infinite, the shadow of the 

 moon is cylindrical, and the position of the place of observation with reference to this 

 shadow can be fonnd by considering the orthographic projection of the moon's disk and 

 the place upon a plane at right angles to the straight line joining the star with the centre 

 of the eai'th. 



<// being the geocentric latitude and p the earth's central radius at the place, the 

 observer moves with the diurnal motion of the earth in a circle whose radius is p cos i//, 

 and whose centre is at a distance p sin (!>' from the earth's centre. The motion in the circle 

 is uniform at the rate of 15° in one sidereal hour, or 15°. 04 in one hour of mean time. 



Projected on the plane perpendicular to the line joining the star with the earth's 

 centre, this circle becomes an ellipse whose major axis is p cos ^' and minor axis p cos ^', 

 sin Ô S being the star's declination. Points on the circumference of the circle are pro- 

 jected into the points where ordinates to the major axis meet the ellipse. The hour angle 

 of the star at any instant being given, the place of the observer at that time will be that 

 point of the ellipse whose eccentric angle, measured from the minor axis, is that hour angle. 



In the American Ephemeris, the hour angle of the star at the time of geocentric con- 

 junction in R. A. is given (in the column headed H). This hour angle, corrected for 

 difference of longitude between "Washington and the place, and applied as above, locates, 

 the observer at the time of conjunction. His place on the ellipse at any mean time-interval 

 before or after conjunction is found by increasing or decreasing the eccentric angle by 

 16". 04 per hour. (The mean solar hour is the time unit of the tabular quantities x', y', in 

 the Ephemeris). 



Measuring from the centre of the ellipse a distance p sin i// cos S along the minor axis, 

 we have the centre of the earth, which is the origin of coordinates for the moon's place. 

 The axis of y is in the direction of the minor axis of the ellipse and the axis of x perpen- 

 dicular thereto. 



The coordinates of the moon's centre at conjunction are and Y, X being equal to 

 at conjunction, and Y being given in the Ephemeris. The qiiantities x' and y' are the 

 hourly changes of X and Y ; (for purposes of prediction x' and y' may be considered as 

 constant, the moon moving approximately in a straight line with uniform velocity). 



Thus the coordinates of the moon's centre are : — 



At conjunction . . . Q and Y 



At one hour after conjunction - x and Y-\- y' 



Plotting these two points and joining them by a straight line, we have the track of the 

 moon's centre during the hour following conjunction. Proportional parts of this length 

 laid otF along this line give the moon's place at any required interval from conjunction. 



Sec. Ill, 1888. 3. 



