Trigonometrical Survey of England. 369' 



and co was correctly derived from these quantities without the 

 employment of any other quantity. But in the assumption of 

 the correctness of these quantities, and by the use made of 

 them in the further calculations, the ellipticity of the spheroid 

 to which these quantities belong was implied. This is dis- 

 guised by the introduction of the length of the geodetical line, 

 but the same ellipticity may be obtained as accurately without 

 this line. From the equation of the geodetical line it is easily 



proved, that making tang* = (J~, )* and tang \\ = (^7) 



. e 7 sin (^'-^) 



we nave - — - = -. : — — -. k~^~ -. — ; — r. 



1 — e i cos y . cos y . sin (wt— m') sin {in -[ m ) 



Now the calculations of the Survey lead to the value e con- 

 tained in this equation, and, as conducted, could not possibly 

 lead to any other. Not having logarithmic tables to more 

 than seven decimals at hand, I cannot determine the angles 

 \J/ and * as accurately as it might here be required. I find 

 from the data given in the Survey the following quantities ; 

 * = 67° 46' 45"'37, V a 67° 46' 47"*37, log e* = 8*1252235, 



and the ellipticity about tt^t? nearly the same as in the Sur- 

 vey. Next in the spherical triangle co = 1° 26' 47"'93 as in 

 the Survey, ^ a 96° 58' 23"-27, ft' a 81° 54' 27"*82, |3 = 

 0° 55' 28"*52, consequently jtt — m = 2' 25"*27; and as a proof 

 that these values are correct, and that the quantities of the Sur- 

 vey belong to the spheroid here deduced, it will be found that 



2e 9 sin X ~ X . cos - *T A ' ■ sin m . cos I. (\ + ~ sin X sin X' j 



= 2' 24"-78. The small diiFerence 0"*49 arising from the im- 

 perfect determination of (*'— 4>) above stated, shosvs that the 

 excentricity is a little greater than the one I arrived at ; namely, 



about fi (p — m being nearly proportional to the excentri- 

 city). 



It will, therefore, be clear that the values used in the Tri- 

 gonometrical Survey belong to a spheroid of an ellipticity equal 



to about — and to no other, and that the corresponding dif- 

 ference of longitude was correctly derived. The employment 

 of the geodetical line gave only the linear dimension of the 

 spheroid, the figure of which was determined without it. But 

 it will be seen how much this figure will change by a slight 



alteration of the data used for finding it. As e°~ is about — , the 



numerator of the fraction r— is to its denominator nearly as 



New Series. Vol. 4. No. 23. Nov. 1 828. 3 B 1 : 74, 



