GEODESIC INVESTIGATIONS. 1G9 



* The solution given to this problem in that -work, as the one to be preferred 

 ■when great accuracy is required, is based on defective conceptions and 

 incorrect formulae. 



In order to obtain the diiFerence of longitude it has been the first object to 

 find, in terms of the given data, an expression for the difference between the 

 angle I),, and the azimuth A!'-, so as to have the side P^S, = ^^ , the angle 

 jPS^ D^, = A!, and the angle D,, of the triangle TS, D„ . 



The general expression given for this difference, which has been denoted by 

 ^, is, when put in the notation adopted in this paper, 



C= 4 , ' a . COS" I', sin 2 J', t} 



and if the given data were I", A", s, 2, it would be 



C= i — ^ cos2 I", sin 2 A". z,f. 

 1 — e- 



And it is assumed that in all cases, the azimuth is less than the angle D, 

 viz., that A' is less tlian D„ and that A" is less than J),, 



Now I have clearly proved (in article 2 of the investigations) that when 

 the latitude I' is greater than the latitude I", 



A' is less than D, 



A" is greater than Z)„ . 

 So that if the magnitude of Q were even admitted to be correctly deter- 

 mined by the above formula, the fact of its misapplication would still 

 cause greater errors in the results of actual calculations, than could arise from 

 neglecting the small angle ^ altogether, and thiis regarding the figure of the 

 earth to be a perfect sphere. 



But a little consideration will suffice to show that the formula does not 

 give a correct value for the magnitude of angle C- ^ov, assuming the 

 latitude I' of the station S' to be 50°, the difi'erence of longitude co = 1°, and 

 the azimuth A' = 90°, it is obvious that the azimuth A" must be less than 90° ; 

 and .•. since 2A' = 180°, one of the values of C given by the formula is zero 

 or = 0, and the other value a positive quantity. 



Now, from the investigations in this paper, it is evident that, by retaining 

 the above-mentioned values of co and V, we might have the azimuth A' less 

 than 90°, and also the tmequal azimnth A" less than 90° ; and that the 

 formula in such case would lead us to infer that the angle f should^be positive 

 whether D, or D^, may be the angle we wish to find. 



It is stated, on page 248 of the above-mentioned work, that the angle f is 

 so small as not to amount to --co of a second even when the geodesic arc 

 between the stations is 100 miles with an qjverage azimuth of 45°. Never- 

 theless, if the angle D,, were to be assumed equal to A" , ami that we were to 

 compute the co-latitude l,, from the triangle, we should obtain a value = L,„ 

 leaving a defect = 5„. Now 5,,. sin A" is very nearly equal the angle A 

 between the normal-chordal planes ; and for an azimuth A' of about 47°, 

 latitude V = 49° 30', and arc s = 60 miles the angle A = 45 seconds in 

 the example worked out in my last paper. Hence 4". 5 ~ sin 47° gives very 

 nearly 6" as the error in the latitude I" if calculated from such data. 



* In the first solution there given, it is assumed that z, = -^ , 2„ = A which ia cquiva- 



lent to assuming that the earth is a perfect sphere. In reality -1- or -£- is always less than 

 2; and it may he seen in note (5), at the end of this paper, that in some instances z,, is 

 greater than 2. However, that method cf solution is tacitly admitted as not suitable 

 when "at accuraov is reaij'red. 



