190 Mr. Ivory on Measurements on the Earth's Surface 



Referring to this Journal, July last, pp. 8 and 9, for the ex- 

 planation of the symbols employed, I shall now put 



Sin4 = £ 



2 2a 



Si„i-= i-*/ (i >-K)« + (?-0°: 

 and it is obvious that sin — • is no other than half the chord 



i 



of the elliptical meridian comprehended between the latitudes 

 of the two stations, which may be computed to any required 

 degree of exactness. If now we substitute these values in the 

 equation at the top of p. 9, we shall get, 



Sin 3 —- -sin 9 — -pusin 2 — ; 



and hence, 



2 cos X cos X \ 2 2 / 



r, in logarithms, 



/ sin — ~ sin ^~- \ 

 Log sin -„- = \ log V * f_ J 



o 2 * . o V cos X cos X' J 



cos X cos X' Jr 2 



(sin 2 X + sin 2 A') (A) 



This formula, when i = 0, coincides with the usual rule for 

 finding an angle of a' spherical triangle when the three sides 

 are given, £ being the base, and I the difference of the sides. 

 If we observe that small arcs of the elliptical meridian and of 

 the equator, which are equal in length, have very nearly equal 

 chords, we shall readily obtain this formula for finding % which 

 is very convenient in practice, viz. 



8 = (x-V) . \ 1 - 1 (-i + 4 cos (*> *')) } ( a ) 



I shall now add another formula for finding the difference 

 of longitude when there is given, the azimuth at one station, 

 and the latitude of the other, together with the length of the 

 chord between them. Let m denote the azimuth at the first 

 station, that is, the angle between the meridian and the second 

 station ; A f , the latitude of the second station ; and y, the length 

 of the chord : further put R for the radius of a sphere the 

 surface of which passes through the two stations, and touches 

 the horizon of the first : then the difference of longitude co, 



will 



