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LXXI. On the Method in the Trigonometrical Sumey for finding 

 the Difference of Longitude of two Stations very little different 

 in Latitude. By J. Ivory, Esq. M.A. F.R.S. $c* 



TN this Journal for October last, I endeavoured to prove 

 * that the method in the Trigonometrical Survey for finding 

 the difference of longitude of two stations, not much different 

 in latitude, was insufficient, and led to erroneous results. The 

 principle of the method is this, That the latitudes being the 

 same, the difference of longitude is independent of the excen- 

 tricity,or it is the same on the surface of a sphere and a spheroid 

 of small oblateness; which, in reality, is consistent neither 

 with experience, nor with other methods of investigation of 

 undoubted accuracy. We are told indeed, that the incorrect- 

 ness of the method, and its want of success in practice, is now 

 allowed on all hands; but the date at which the delusion was 

 dissipated is not mentioned. As my intention was merely to 

 overturn an insufficient method of calculation, not to establish 

 a new rule, I neglected such small quantities as could not be 

 distinguished from the unavoidable errors of observation. 

 Thus the small quantity in the value of /3, which I neglected 

 (p. 243), would be produced by a small variation in the length 

 of the chord y, amounting to about 18 feet in the distance 

 between Beachy Head and Dunnose, more than 65 miles. 

 Assuredly a method of calculation which requires such nice 

 accuracy in the data of observation is not a very solid founda- 

 tion on which to place any conclusion respecting the figure 

 of the earth. There is also an omission in the equations (#) 

 (p. 243), arising from supposing R = a (p. 242), which, how- 

 ever, will affect the azimuths only a small fraction of a second 

 in the ordinary circumstances of the problem ; that is, when 

 one azimuth is greater, and the other less, than 90°. In the 

 extreme case, of which there is no instance that I know of, 

 when the azimuths are both less than 90°, and nearly equal, 

 the last-mentioned omission will affect the accuracy of my 

 formula, because the quantity neglected does not vanish when 

 the two latitudes become equal. There is, however, no doubt 

 that, in the ordinary circumstances of the problem, my me- 

 thod, which takes the excentricity into account, comes nearer 

 the truth than the method in the Survey, which entirely neg- 

 lects the figure of the earth ; and this seemed sufficient to 

 answer my purpose. But it is not easy to put down an autho- 

 rized error; although I shall now attempt to accomplish 

 this task by new investigations, to which it will be impossible 

 to object. 



* Communicated by the Author. 



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