Mr Galbraith's Barometric Measurements of Heights. 317 

 might prove a little troublesome to persons not very conversant 

 with such calculations, and the use of the centesimal thermome- 

 ter also, which it requires, is not very general in this country, it 

 appears that a formula, or rule deduced from it, dependmg up- 

 on calculations of an easy nature, and adapted to Fahrenheit's 

 thermometer, avoiding the tedious process of obtaining the cor- 

 rection for the mean temperature of the air employed by Roy, 

 &e. would be useful to traveUers who might not have access to 

 tables, or when the operation is performed both ways, the one 

 might be a check upon the other. 



General Investigation. 

 In most works which treat of the properties of logarithms, it is shown that 



Now, if }^ = f ; then n = |=| ; whence by substitution, 



in which B expresses the height of the barometer in inches at the lower sta- 



tion, b that at the upper, M is the logarithmic modulus, and consequently 



2 M =0.868589. 



. In order to simplify, let ?- = «, |^ = P, and t M = m ; then 



log« = m{/3+i/5'+^/3= + &c.}=«(3(l + M^ + ^/5* + &c.) . • • (3.) 



To abridge, let 1 + J P'^ + I /B" + &c. =s, and equation (3) becomes 



log a ==nifis . ,., 



If . denote the expansion of air for F of Fahrenheit's thermometer, in which 

 the freezing point is at 32', then formula (4) must be multiplied by 

 l+e{-^/-Sr)=l+%{t + l'-6i') = l-32e + %{t + f) . ■ • (5.) 

 in which t and t' are the temperatures of the air at the bottom and top, by 

 the detached thermometers. ,.,,„„ ,„„«f 



Let c = 60155 English feet, the factor nearly constant by which log « must 

 be multiplied at 32" to convert it into English feet, then log «xc = H the 

 height in feet ; consequently at any other temperatures of which the mean 



t + t' 

 is — g— 



n = cmr^s{l—S2e + %{t + n} ^^'^ 



This may be put in a different form :— 



ll^cm{lS2e+e(,t^t')}^s C-) 



Now c X m = 601 55 X 0.868589 = 52250 feet. ^ ^ ^ ^^^^^ 



According to Roy ' ^^^^^^^ 



Laplace, 



Dcluc and Saussure, . . • • O-Q^^'J-i 



The mean of these gives 0.00230 



