58 isr. F. DUPUIS 



the moou and the siiu, upon the surface of the heavens, a small portion of which is taken 

 as a plane, taking the place of observation as the centre of projection. This method gives 

 us matters in their natural appearances ; and in the projection, the star becomes a point 

 on the paper, and the moon becomes a circular disc which moves over and obscures or 

 occults the star ; or in the case of an eclipse, we have two discs, representing the ' §• and 

 3, of which the sun remains fixed, while the moon, moving along her path, passes over 

 and obscures more or less of the sun. 



If we were situated at the earth's centre, nothing would be easier than, from the 

 Nautical Almanac, to lay down upon paper the position of a star to be occulted, and the 

 positions of the moon's centre from time to time at intervals of an hour, or a half-hour, or 

 ten minutes, or less ; and the moon's apparent motion during these intervals would be 

 sufficiently irniform to admit of the employment of proportional parts in determining 

 smaller intervals. But from our position upon the earth's surface, the moon suflFers 

 parallactic displacement ; and this displacement is practically the same as that of the place 

 of observation, with respect to the earth's centi-e, as seen from the centre of the moon. 



Now putting /T to denote the moon's horizontal parallax (i.e. the angular value of 

 the earth's radius as seen from the moon), â to denote the moon's declination, Ç) the 

 geocentric latitude of the place of observation, and a the hour angle of the moon east or 

 west of the meridian of the place, we have for the displacement, from the well known 

 formulae for the transformation of spherical coordinates : — 



D = 7T sin qj cos Ô — tt cos cp sin â cos a 

 J ^ TT COS cp sin a 



where D is the displacement of the moon's centre, north if + and south if — ; and -1 is 

 the displacement of the centre in E.. A., east or west from the meridian of the star. 



To find these c[uantities : — 



In the application cp is a fixed angle for any particular place, and tt, â, a, are found 

 in, or through the means of the Nautical Almanac for any particular epoch. 



These being obtained draw any line QOS and OG a perpendicular to it. Make the 

 angle GOP equal to <p, the latitude of the place of observation, and the angle GOII equal 

 to the declination of the moon at mean time of conjunction, as taken from the Nautical 

 Almanac. Next, take OP equal to tt from a scale of equal parts. The value of ^r is best 

 expressed in minutes of arc, and all other lengths concerned must be reduced to the same 

 unit and taken from the same scale of equal parts. Draw PQ perpendicular to OQ, and 

 with Q as centre, and QP as radius, describe the quadrant PE, and take the augle PQA 

 equal to a, the hour augle of the moon for the particular epoch required, i.e, the angle 

 between the meridian of the moon and the meridian of the place of observation. Draw 

 Aa perpendicular upon QP, and aa' perpendicular upon OH. 



Then for the hour augle a, aa' is ec[ual to D and Aa is equal to J. 



To show this. 



aa' = QO cos ô — Qa sin 6. 

 But 



QO = TT sin qj, and Qa = QA cos a ^ n cos <p cos a 



. • . aa' = 71 sin cp cos ô — tt cos cp cos a sin d r= Z>, 

 and 



Aa = QA sia a = ;r cos ^ sin a =: J 



