648 PHILOSOPHICAL TRANSACTIONS. [aNNO 1795. 



tained correctly, the difFerence of longitude between Beachy Head and Dunnose, 

 as thus found, must be very nearly true; since the difference between the sums 

 of the angles which would be observed on a spheroid and those on a sphere, 

 having the latitudes and the difFerence of longitude the same on both figures as 

 those places, is so small as scarcely to be computed: and it is easy to perceive, 

 that the distance between the parallels is obtained sufficiently correct, since an 

 error of 1 5 or 20 feet in that meridional arc, will vary the length of the degree 

 of the great circle but a very small quantity. 



It may possibly be imagined, that because the vertical planes at Dunnose and 

 Beachy Head do not coincide, but intersect each other in the right line joining 

 these stations, neither of the two included arcs is the proper distance between 

 them, and that the nearest distance on the surface must fall between these arcs; 

 but it is easy to show, that in the present case, the difFerence must be almost 

 insensible. In fig. \g, let b be Beachy Head, and ebp its meridian, and n and 

 M, the points where the verticals from Beachy Head and Dunnose respectively 

 meet the axis pp. Now it is known, that if the planes of two circles cut each 

 other, the angle of inclination is that formed by their diameters drawn through 

 the middle of the chord, which is the line of intersection. Therefore, if we 

 draw BM, and also conceive d to be Dunnose, and ep its meridian, and join dn; 

 it is evident, that either of the angles nbm, ndm will be the inclination of the 

 planes very nearly, because of the short distance between the stations, and their 

 small difFerence in latitude. In the ellipsoid we have adopted, the distance mx 

 is about 62 fathoms, and hence the angle nbm, or ndm, will be found between 

 2 and 3". The value of the arc between the stations is about 55' 30", and its 

 length 339401 feet; hence the versed sine of half the arc will be 685 feet nearly; 

 now suppose the versed sines to form an angle of 3", the greatest distance of the 

 vertical planes on the earth's surface between the stations, will be but about -^ 

 of an inch. It may also be remarked, that the inclination here determined, is 

 the angle in which the vertical plane at one station cuts the vertical at the other; 

 and therefore no sensible variation can arise in the horizontal angles, on account 

 of the different heights of the stations. 



If the figure of the earth be that of an ellipsoid (fig. 20) then br, which is 

 perpendicular to the surface at the point b, is the radius of curvature of the great 

 circle, perpendicular to the meridian at that point; therefore the length of a 

 degree of longitude is obtained by the proportion of the radius to the cosine of 

 the latitude. Thus at Beachy Head, where the length of the degree of a great 

 circle is 61 183 fathoms nearly, we have this proportion: rad. : cosine 50° 44' 24* 

 ::6i 183 : 38718 fathoms, for the length of the degree of longitude. And at 

 Dunnose, as rad. : cosine 5(f 37' 7" :: 61 182 : 3881 8 fathoms for the length of 

 the degree of longitude, being about 100 different from the former. But nearly 



