68 GEODESIC IXVESTIGATIOXS. 



latitudes, azimutlis, angles of normal planes, &c., have been 

 obtained as cori-ectly as the sides of tlie triangles in the most 

 accurate trigonometrical surveys. 



The second test applied to the correctness of the work would be 

 sufficient to show the calculated results to be close approximates to 

 rigorous accuracy, had the method of solution itself not furnished 

 an easier and more rigorous test. 



I may mention that Dalby's Theorem is retained in " The 

 Account of the Principal Triangulation of Grreat Britain and 

 Ireland," and submitted as a most imjjortant formula, with two 

 proofs of its approximation to truth, and historical notes of a 

 curious character. 



It is there stated that, in applying it to the determination of 

 the diiference of longitude of Beachy Head and Dunnose, it gave 

 „ results at variance with those obtained otherwise. 



That it would be likely to give such a result in that case is 

 evident — for when the deviations of the plumb-lines at the stations 

 is such as to affect the azimuthal angles separately in a greater 

 degree than it affects the sum of the azimuths, the like paradox 

 will present itself, whether the latitudes have been correctly 

 obtained or not. However, Dalby's theorem is a close test to the 

 accuracy of calculations for longitude, made by using correct 

 data. 



4°. From the example in which d is assumed as 60 miles, it 

 appears that the a between the " normal chordal planes " is 

 4'' "491. And assuming one of the stations to be 4,000 feet higher 

 than the other ; then since 4,000 multiplied by the sine of 4""491 

 r=:"087 feet, and that this fraction of a foot subtends but a very 

 small fraction of a second, we may consider the traces of the 

 "normal chordal planes" on the earth's surface as one and the 

 same trace. However, this small angle causes the shortest dis- 

 tance of the normals from each other to be 455'78 feetzz | \ (p'+p") 

 — versin d]' sin a. 



5°. The method of investigation is no doubt of a very elemen- 

 tary character when compared with the analysis usually employed 

 in treating such questions ; but, for this reason, it aff'ords a clear 

 view of the various relations of the involved entities, and leads to 

 more elegant formulae (as a natural consequence) which can be 

 worked with great precision. I need scarcely mention that the 

 utmost precision is absolutely necessar}^ : for if the relative 

 latitudes, longitudes and azimuths of the principal stations of a 

 trigonometrical survey be not attainable with at least equal pre- 

 cision to that of the lengths of the sides of the triangles, their 

 positions cannot be geographically defined with accuracy. This 

 will appear evident when we consider that an error of one second 



