'He Further Remarks on the new Method of determining the 

 tan 2^ — p + ~ -V i^c. 

 cos ;^ = 1 — -- + &c. 



tan3 « 



ti = tan u — - + oLC. 



whence, by rejecting all the powers above the thu-d, as of no 

 value (which, in the case of the pole-star, may be safely done) 

 and substituting the series in the general expression above 

 mentioned, we have 



tan u = cos t 



0' + 4) 



(«3 \ cos' t. ]y> 

 P+ —) 5— 



= j;. cos ^ + -^ (cos t — cos"* t) 



= p. cos f -{- -^. sm- 1. cos t. 



I)''-, cos 'J t 

 cos U = I — - — 



cos M , p'i P'^ q , 



= 1 4- -i '—. cos' t 



cos;> ^2 2 



=> 1 + -f. sin- 1 

 cos (4' — ?0 = cos z -{- -^. sm- /. cos z 



= cos ;:; + -|- . sin'' /. cot z. sin ;:; 



(\J; — ») = S — -^. shl'^ /. cot Z 



4/ = 2 ^ sin- 1. cot X +p. cos<4- -^rsin^ /. cos t 



— z + Ip + C) cost — B cotz. 

 which is the very same formula as that which I have given in 

 my former ^^aper. 



This formula is an approximate one ; but, as far as the pole- 

 star is concerned, is correct : the terms which are omitted not 

 affecting the result in any sensible manner. If the general 

 formula be used, it may be applied with equal advantage to 

 8 Ursre Minoris and other stars not far distant from the pole : 

 and thus the means oi' deducing the latitude from sxich obser- 

 vations will be multiplied. But if that formula be converted 

 into series, we must (in these cases) retain the fourth and some- 

 times the fifth powers of p : which however will not render 

 the approximate formula more intricate or laborious, and the 

 computer may adopt it in lieu of the general one. Yet I doubt 



the 



