perpendicular to the Meridian. 191 



will be found by this formula, which is exact and easily de- 

 monstrated by the most simple geometry, viz. 



a cos 7J 4 Ra 



Now it is obvious that R will always be very nearly equal to 

 a ; and since y is always a small part of R, or of a 9 we may 



take ~r as equivalent to ■—. But if we make sin-£- = — . 



4a 2 * 4R* 2 2a > 



then cos ~ = V 1 — 5 and — V 1— -£r = 2 sin -^- 



2 4« a a 4a'* 2 



cos — = sin /3 : and thus we obtain, 



cv sin /3 sin m ,- : — - — 7 



Smw= -. — Vl— 2esin 2 A', 



COSX' 7 



or, in logarithms, 



Logsin» = log(^^)-M s sin'V. (B) 



In illustration of these rules I am tempted to apply them 

 for finding the difference of longitude between the observa- 

 tories at Greenwich and Paris. In the new survey the length 

 of the arc drawn from Dover perpendicular to the meridian 

 of Greenwich, is 50634? fathoms*. I consider the foot of this 

 arc as the first station in the formula (B), and Dover as the 

 second station. Hence, m — 90°; X' = 50° 7' 45"-6. As the 

 given distance is not a chord, but an arc on the earth's surface, 

 we shall find /3 by reducing the given length, taken as an arc 



of the earth's equator, to degrees : therefore $ = 6 x 3600" 



= 49' 53"*3. The formula (B) will now give us the longi- 

 tude of Dover equal to 



1° 19'23"'78. 



As we have no azimuth either at Dover or Dunkirk, we must 

 apply the formula (A). The two latitudes are, 



Dover A = 51° 7' 45"-6 



Dunkirk A' = 51 2 8-5. 



General Roy makes the distance from Dover to Dunkirk equal 

 to 244916 feet f, or 40822 imperial fathoms. According to 

 the mode of calculation in the Survey, this length is not a chord, 



but an arc on the earth's surface ; and hence /3 ss ^-r x 3600" 



=40' 14"-87. The formula (a) gives 8 = 5' 36"-91. We have 

 therefore, by the formula (A), the difference of longitude be- 

 tween Dover and Dunkirk equal to 

 1° 3' 19"-10. 



* Phil. Trans. 1828, p. 180. f Trig. Survey, vol. i. p. 147. 



The 



