Astronomical and Nautical Collections. 147 



already explained, >/« dr = - p / ' ^- (1 + . 414214 mp 



+ .269649 m°-p°- + . 200865 mV + ...) or, taking — s*= 



P 

 3403, and m = 798, r = .010423 (1 + .097133 + .014830 

 + .00259 + [.0005]) = .010423 X 1.1151 £2 39' 57"; so 

 that the former result was too great by 18": and if we make 



JL == 3540, and m = 766, we find/- = .0097988 (1 + .08963 



V 



+ .01263+ .00203 + [.00040]) == .0097988(1.1047) = 

 .010825 == 37' 13", or 16" less than the former computation 

 made it; the difference, which before came out 2' 46", being 

 now found, a little more accurately, 2' 44". 



The relation between x and z may be computed from the hy- 

 pothesis of an equable variation of temperature in ascending, 

 according to the statement expressed by the equation z ~ y 

 (1 + tx — t) (Coll. VI. 7. d), or z = yiv, whence 



i dz zdw dz dw tody , 



w ww z w z 



dy = — mzdx, and — sd — mwdx ; consequently 



z w 



hlz = hlw — mfwdx, and z = wer- m f wdx = ( 1 + to — t) 

 e —m(x+tfxx—n) . an( j from this expression we may find the den- 

 sity z corresponding to any height x, upon the supposition that 

 the bulk of a given quantity of air varies proportionally with a 

 uniform variation of temperature, and not uniformly, as the ex- 

 periments of Schmidt and Gay Lussac induced them to infer 

 with respect to ordinary temperatures. (See Nat. Philos. 

 Vol. II, p. 393.) If we computed the horizontal refraction 



dz 

 from this equation by means of the series beginning with — , 



we should have to substitute, for dz, (1 + tx— t) e -m(T+it.T>-<<) 

 (—mdx — mtxdx) and for s, x— 1. 



Besides the equation y = az' z + bz 3 + cz 4, + . . . , there 

 may probably be many others, not far from the true constitution 

 of the atmosphere, which would afford finite expressions for the 



L 2 



