from Measurements of the Meridian, 433 



may thence deduce its curvature at certain fixed points ; and, 

 in order to judge more accurately of the near agreement of 

 the measurements with the theory, we may compare them se- 

 parately with the curvature of that portion of the meridian to 

 which they are nearest in situation. For this purpose I shall 

 use the notation D (a) to denote the length of a degree of 

 which the middle point is placed in the latitude A. The ex- 

 treme latitudes of the degree will therefore be A + ^ and A — \ ; 

 and, in the formula (A)*, we shall have n = 1°, p sin n = 1°, 

 cos n = I, m = 2 A; consequently, 



D(A) =A{1- S (i + f cos 2 A) % ? ( T V + U COS4A)}. (C) 



In this expression put A successively equal to 0°, 45°, 90° ; and 

 we shall have, for the length of 1 °, 



Fathoms. 

 At the equator, D(0°) = A(l - 2e + s 9 ) = 60462-4 



At latitude 45°, D (45°) = A(l -J i - £s 2 ) = 60757 

 At the pole, D (90°) = A(l + « + e 2 ) = 61054 

 Let us now compare the arcs measured in Peru and India, 

 with the curvature of the meridian at the equator. From the 

 expressions just set down, we get, 



A=D(0°).(1 + 2s + 3 e 2 ); 

 and if we substitute this value in the two measurements men- 

 tioned, we shall get these three equations, the last being the 

 sum of the other two, viz. 



188510 = D(0°) (3-11750 + 0*0089 . s -0*005. s 2 ) 

 598630 = D (0°) (9-89589 + 1*5933 . s-0'546 . s 2 ) 

 787140 = D(0°) (13*01339 + 1*6022 . s-0*551 . s 8 ) 

 In the first of these expressions the terms containing g may be 

 neglected, and D (0°) will be found equal to 60468, or near 

 6 fathoms too long, which arises from the Peruvian arc 

 being about 20 fathoms too long. If we substitute the value 

 of s in the two remaining expressions, we shall get 60461*3 

 and 60462*9 for the value of D (0°), hardly different from the 

 length previously deduced from the two elements. 



The French arc is nearly bisected by the parallel of 45°, 

 and it must be compared with the curvature at that latitude. 



NoW ' • A = D(45°).(1 +i + § s *); 



and, this value being substituted in the expression of the mea- 

 surement in question, we shall obtain, 



751567 = D (45°) . (12*3703-0*0960 . i + 0*310 «-). 



* Phil. Mag. and Annals for May, p. 344. 

 New Series. Vol. 3. No. 18. June 182S. 3 K Here 



