66 GEODESIC INVESTIGATIONS. 



Notes. 



Assuming tlie earth to be an oblate spheroid, of which the axis 

 of revolution is the polar diameter, and that we have the values 

 of «, h, its polar and equatorial radii, and those of the given data 

 with all attainable accuracy, I propose to show the assumptions 

 on which the investigations are based, to be compatible with the 

 most rigorous requirements in the actual practice of trigonome- 

 trical surveying. 



1°. First, then, I may observe that when the geodesic arc d is 

 not over 60 miles in length the approximate values of Z" and A" 

 derived from the formulae recommended by Oliver Byrne, C.E., 

 in his treatise ou Greodesy in "Bohn's Dictionary of Engineering" 

 (which I have used in preference to all others, as giving the 

 nearest approximations hitherto attainable), are equal in every 

 respect to the absolutely rigorous values in their applications to 

 the determination of 2 and c the circular measure and chord of 

 the geodesic arc d, and that this would be the case even were 

 their differences from the true values of I" and A!' twice as large 

 as they really are. We can easily prove this when the stations 

 S', S" are ou one meridian, by means of plane analytic geometry ; 

 but when, as is generaUy the case, the stations are not so situated, 

 it is necessary to elucidate the relations which subsist between 

 radii of curvature on the spheroidal surface of the earth. This 

 may perhaps be effected by the following observations, without 

 entering into the extensive calculations necessary in order to 

 afford a rigorous test. Although any two normals to the 

 spheroidal figure of the earth will cut each other only when the 

 stations are either both on one meridian or on a parallel of 



latitude, it can nevertheless be clearly inferred that , , „ is an 



extremely close approximate to the circular measure 2 of a 

 geodesic arc d not over 60 miles in length, and the more so the 

 greater the difference in longitudes of the stations S', S". In 

 the case in which d is part of a meridian, the angle made -by 

 the normals or radii of curvature at its extremities is at once 

 attainable as the difference of the geographic latitudes of the 

 extremities. But when the geodesic arc d is not a portion of 

 one meridian, we have a means of computing the rigorously 

 correct value of 2 supplied ^to us by the higher calculus. We 

 can form the equation of the spheroidal surface of the earth 

 referred to rectangular axes, making the centre the origin and 

 the polar diameter the axis of z, and find expressions for the 

 co-ordinates of the stations S', S" in terms of their difterences of 

 latitude and longitude and the equatorial and polar radii. We 

 can also form the difterential equations expressing the circular 



