MR EDWARD SANG ON THE DEFLECTION OF THE PLUMMET. 91 



therefore the deflection of the plummet due to the sun's attraction may be 

 stated as 



1" 



j^ . cos 2a (O). 



The moon, though much nearer to us than the sun, is yet so small in com- 

 parison, that the effect of a pressure equal to her attraction would only derange 

 the plummet through one second. In this case, however, the ratio of D to R is 

 only as 60 : 1, and therefore the deflection due to the moon's influence is 



gQ- . sin 2« ( D ) 



or three times that due to the sun. 



These expressions represent almost exactly the deflections corresponding to 

 the mean distances of the two luminaries ; they would, if we were to seek for 

 extreme precision, need to be varied in proportion to the third power of the hori- 

 zontal parallax. 



The deflection of the plummet takes place in the vertical plane passing through 

 the attracting body ; and the actual position of the apparent nadir point may be 

 obtained by referring it to co-ordinate axes drawn, the one from north to south, 

 the other from east to west through the true nadir. 



Putting X for the ordinate measured southwards, z/ for that measured west- 

 wards, A for the north latitude, 8 for the north declination, and h for the hour 

 angle, we easily obtain 



« = — sin 2x . sin 3^ — cos 2x . sin 2d . cos h + sin 2x . cos 5^ . cos K' 

 y= +sin X . sin 23 . sin h + cos X . cos 3^ . sin 27t, 



the values being in terms of the maximum deflection assumed as unit. 



These form the equation of an elliptic epicycloid produced by carrying the 

 centre of one ellipse round the circumference of another, the periodic times being 

 24'' and 12'' respectively, and the motions being such as to generate areas uni- 

 formly round the centres. Like all epicycloids, these curves change their appear- 

 ance with every change in the constants ; they have distinct phases, according as 

 the observatory is within the arctic circle, above latitude 45°, on the parallel of 

 45°, below latitude 45°, intertropical, or on the equator. 



These phases are figured in the accompanying drawings, in which the hour 

 angles are marked from the meridian ; and which explain, better than any words 

 can, the remarkable changes which the motions of the nadir point undergo as 



1" 

 the attracting body changes its declination. The line AB represents ^^ for the 



1" 

 lunar deflection, and jgg for the solar. It may be well to advert to some of the 



leading features. 



When the observatory is within the arctic ch'cle, and when the attracting 



