106 



ON THE FORMULA FOR CALCULATING, &c. 



The nature of this last substitution, has been shewn in my work on 

 the measurement of an Arc of the Meridian (Page 61) and is simply thus. 



d (hy. log (1 + ^)) =: — +dx.(\lfx-{-x^'+x^-\- &c.) 



* ' x^ x^ x^ 



... hy.log(l+x)= + + y + y + — -f &c.) 



J^v' Log(l+a;)::^ +iSf^.(l + | +^ + ^&c.) . 



in which, w^hen x is very small, the series converge so rapidly that all 

 terms but the first may be omitted, and we get merely 

 'Log. {\ ±x) — ±M X. 



Therefore Log. (1 + cot P. tan h P) — ± M. cot P. tan 5 P. 

 Taking now the elements as in the first of the above instances, viz. 



PS = 1° 36' C''; h F — SO'; P — 89° 17' 14'''.8 ; X = 24" 0' 0^ 

 the computation will be as follows : 

 To find M. cot P. tan h P 

 9.63778 



9.11943 Tan. 7° 30' 

 8.09472 cot P 



6-85193 Log, o f + 0.0007111 



K 2 PS =: 3° 12' 0'^ . . Log. Sin 8.7468015 



i:: . I- a P = 3° 45' 0^^ . 2 Log. Sm. . . 7.6311970 



' P Log. Cosec 0.0000336 



X = 24° 0' 0''' Log. Sec 0.0392693 



Ar. Co. Log. SinV 5.3144251 



a Z above 54''''. 005 . 1.7324.381 



a Zbelow 53^ 82 9 1.7 310159 



The above computations will shew that the approximate method may 

 be quite as much relied on as the more elaborate one, and it will appear 

 on pursuing the enquiry that, for about thirty-two minutes prior and an 

 equal lapse subsequent to the maximum, the Polar Star only varies one 

 minute of space in Azimuth in the latitude of 24° 0' C''. 



