142 Astronomical and Nautical Collections. 



be admitted by natural philosophers, in general, as applicable 

 to the phenomena in their whole extent. 



Example B. We may also obtain a finite inverse series, nearly 

 resembling that of the Nautical Almanac, from the equation 

 y = . 57 + . 43z 3 , which is obviously impossible in nature, since 

 it supposes a constant pressure after the density has vanished. 

 A result, however, nearly identical, may be deduced from the 

 supposition 3/ = 3 . \z°~ — 4 . lz 3 + 1 . lz\ which implies an at- 

 mosphere terminating at the height of about 14 miles ; although 

 the series thus obtained would extend to a fifth term, instead of 

 ending at the fourth, but without producing any material 

 difference in the result. Considering, indeed, the analogy be- 

 tween logarithms and high powers, it is not improbable that the 

 true value of y might be very correctly expressed by a series of 

 this form, however complicated it might appear at first sight. 

 The value of x and the height h = 20900000 (r - 1), in feet, 



might be found from the fluxion &x = ~ ^ — 6.8dz — 12.3zdz 



mz 



+ 6.8z=dz, andx-l = — ( - 6 . 8z + 6. 15z* - 2.27?' 3 

 m 



+ 2 . 92), or h sr 27300 (2 . 92 - 6 . 8z + 6 . 15z a - 2 . 27z 3 ) ; 

 which becomes 21300, when the density z is reduced to _;and 



the pressure y — .444. 



Example C. a. As the most unfavourable specimen of the ap- 

 plication of this method, we may take the case of an equable 

 temperature, at the horizon : and first suppose, with Laplace, 



that wi == 798, and _L = 3403, so that —L. r= 4.2624, Z Si 

 p mp 



■ V — 1 = 3.2624, since z is here = y, and f = 1; 

 mp 



7= JL. 1 = 126?!, F 1= 1 Y, Y n = 2y„andF 3 = 

 mp p p p p 



— Y„. Hence we have YZ = , 13 ' 9056 = Z', and the equa- 



P ' P 



tion becomes 1 = 1.6312r' +.5794r'« + .4603r' 3 -f .<£>93r'* 



