&EODESIC INVESTIGATIONS. 67 



measure d-2. between two consecutive normals to any geodesic 

 arc d which is not a meridian (thus assuming the arcs 

 Oi, Os, ... «ii, into which we conceive d divided to be infinitely 

 small, of the Jirst order) ; and we can then integrate between 

 the limits or extremities of d for the total circular measure 2 of 

 the arc d. That we shall, by such means, obtain the exact value 

 of 2 is evident ; and for the method of integrating the differential 

 equations Salmon's " Geometry of Three Dimensions" can be con- 

 sulted. 



From this and the fact that the approximate expression 

 for 2 holds even in the case in which S' and 8" are on one 

 meridian (in which case the difference of curvature of the 

 extremities of c? is a maximum) it should be evident that the 

 greater the difference of longitude of 8' and S" the nearer to 



absolute accuracy will be the expression -rj—, for 2. 



However, if we be content with approximations to accuracy 

 equal to those by which the sides of the triangles have been 

 obtained, and that we wish to keep our results in strict con- 

 formity with the lengths of such sides, then it must be admitted 



2d 



that the va,lue ■ - , „ for 2 is preferable to the absolutely 



rigorous value were such rigorous value to give results whose 

 differences from those obtained by means of the approximate one 

 could be appreciated in practice : but such is not the case. And 

 for like reasons, it is evident that |^ 2 is the proper value for the 

 circular measure of the angle which the chord c of the arc d 

 makes with the tangent plane at either of the stations. 



2°. The angles at the centre of the earth G as obtained from the 

 chord c,and central radii /,r",can be easily proved to be correct to 

 the thousand part of a second, even though r" may have been 

 computed from a value of I" differing by 1" from the correct 

 value. 



3°. The method of solution has a great advantage in supplying 

 a rigorous test to the accuracy of all the calculations, inasmuch 

 as the magnitudes of the three angles of the triangle, whose base 

 is the chord c and opposite vertex the centre of the earth, are 

 found from three different triangles — one of which is the plane 

 triangle itself, and the two others the principal spherical triangles 

 used in obtaining the most important entities. In the example, 

 which I have worked out, it may be seen that the sum of the 

 three angles is about -iho of a second over 180° ; but this is owing 

 to the tables of logs being carried out to seven places of decimals 

 only, when tables of ten places of decimals are necessary in order 

 to find results correctly to ro 0^ part of a second. However, the 

 application of the test is sufficient to prove that the computed 



