Astronomical and Nautical CoUeciions. 363 



example of its application to the hypothesis of Laplace, which 

 is expressed in so intricate a form, as not to be of the most 

 manageable nature, and as to be very liable to some misinter- 

 pretations. 



My series, as it was actually employed for the construction 

 of the table printed in the Nautical Almanac, {Astr. Coll. VIII. 

 ^n.)\sA^vJ--^-[B-L,^y:L+Cv^^ + ^C{B-\ 



s"-') — • and I have demonstrated that A being = p, and ^ = 

 ^y and r = ^, -B must be _^, and C = -L ( -^ + 



^ Y, whatever may be the relations between the density z, 

 and the pressure 2/ : and if we put | = ^, we shall have, still 



more compendiously, C = -^— ^ ^^ + ^ ) ~ 3^^^"^ mJ' 



. d:c _ K" and f - Ml-. 



smce — = ^5 *nQ ^ — • 



(Ir mpsz mpsz 



Now in the hypothesis of Laplace, (Mk. Cil. X. §. 7. P. 264), 

 "M = s - 0,000293876 (1 - J- ); § = i?)V- + «. 661,107] 

 ^-«..jt8,04 ... oj.^ in the symbols of the series, " m" == 1 — 

 L-p{l-z), and z =: (1 + fcw) e"'"; making 661,107 = 



X 



p, and 1348,04 = v. 



Hence, we obtain dw = — + pdz, and dz ^ i^e-^ du - 



XX 



dx 

 vzd«; and making ^ce-"" - ,z - U, dz = t^dw =: U— + 



p mz, dz-pUdz:= U^^, and dz = ^ ^ : and d, 



being = - mzd^, | = ^ = -»nx^ (i-^) =«»^^ (P - 



1 \ /• df/ s •if^. 



-L J ; consequently d? = 2 -i- dx + mx'-jjjj ; - wo; -^^, 



