146 Astronomical and Nautical Collections. 



Now assuming f'=l, r' becomes 1.27, r'" 1=1.65, F=.40 ; 

 and T=: 19.75, instead of 8.0466, that is 11.7 days too much. 

 If we take §'='.5, we have r'=:.99, r"'=:1.14, F=:.l 55, and 

 T=6.15 days, or 1.9 too little. Hence we may conclude, that 

 f' cannot be very different from .56 ; and we find for 

 e'=:.56 e' = .57 



r'= 1.01262 »'=1.01662 



/"= 1.19773 r"'=1.20641 



A"=. 18546 A"-. 19020 



T=8.0121 T=8.2402 



The error of the former supposition is —.0345, the differ- 

 ence of the two values of T .2281 : consequently, the true 

 curtate distance §' is .56151, and we have 



r'=1.0139 r"'=1.1991. 



Now, according to Halley's theory, the true values were 

 r'r=l.0144 and r"'r=1.2000; so that our method gives these 

 distances perfectly correct to the third place of decimals. 



§ 50. 

 These examples are sufficient to show the convenience and 

 conciseness and certainty of the proposed method; upon 

 which I shall make a few further remarks. In order to deter- 

 mine the time from the distances and the chord, we have the 



1 / ^ - \ 



formula T= ^2~U'"''^ r"' + k"'] - — {r'+r'"— K'Y J ; and in 



order to find it the more readily, tables have been computed 



upon these 

 r'-\-r"—li' 



T -\-v"' -\-k" 



upon these principles ; taking B = , and D = 



-, we find the respective times in the tables, and 

 their difference, or, if the angle is greater than 180°, their sum 

 gives the value of T. Such tables are to be found in the Berlin 

 collection, but they are not very accurate : Pingre has im- 

 proved and corrected them, in his Cometographie. 



Since these tables are only calculated to hundredth parts of 

 B and D, I have only found it convenient to employ them 

 when no great accuracy is required, as in the first preliminary 

 experiments with a value of §'. If we wish for great precision, 

 the proportional parts are troublesome, since the first differences 



