J'rom the Vertical to the Earth's Surface. 323 



gled plane triangle, and consequently the amplitude towards 

 the north or south of east may be readily obtained. 



If the heavjf body fall an English mile or 5280 feet in the 

 latitude of Loudon, A will be 2-89 feet only, and sin K x cos A 

 = 0"4872 ; whence 



d = 2-89 X 0-4872 X 00035 = 0*0049 or 0*005 nearly, and 



D= V' 8-3521 + 0-000025 = 2-8900045. Consequently ^/' has 

 almost no effect to increase D, while the direction must like- 

 wise be nearly due east, and therefore both these corrections 

 may in every case be omitted. 



merson gives D = ~ ii) 



in which D is the deviation, d = \/ -^ ; m, the given 



height fallen through ; p, the cosine of the latitude ; r, the 

 radius of the earth; c, 3-141593, the circumference of a circle 

 to diameter unity; t, the time of the rotation of the earth 

 about it saxis; and/ = ^g = 16-1 feet; consequently if the 

 formula be written at full length it becomes 



U = - — , X — — X 7K X P V — :- (2) 



which evidendy is far from being convenient in practice. 



Since the quantity m must in general be very small in com- 

 parison with 3r, it may be neglected without sensible error, 

 and then 



T-v 2 2c / ni 



B=— X— X7nx P\/-j- (3) 



If in this formula we write A for D, h for m, sin 9 for /;, 



2c. 



and n for — , equation (3) becomes 



A = — ?e A sin 9 s/ -^ (4) 



which is the formula of Laplace given in the Bulletin des 

 Sciences, No. 75, and is therefore almost identical with that of 

 Emerson pul)lished in his Alj^cbra many years before. 



It appears that the formula may be very simply obtained in 

 the following manner. Let a he the altitude in feet from which 

 the body is let flill, r the radius of the equator, or even the 

 mean radius of the earth, and tt tlie circumference of a circle 

 to diameter unity. 



'i'he circumtiirence of a circle at the height a is 

 2ir {r -T a) = 2 tt r + 2ir a 

 at the surface Iir x r =. 2 tt r 



Difference, or A = 2 tt « . ... (5) 



the distance which a point at the height a describes more than 

 at the surface during one rotation of the earth. 



2 T 2 By 



