524 PHILOSOPHICAL TRANSACTIONS. [anNO 1774. 



rection, on account of the difference of heat of the barometer at the two stations, 

 may be reduced to a still easier expression, in which the variable quantity b, the 

 height of the upper barometer, shall not appear. The fluxion of a logarithm is 

 to the fluxion of its natural number, as the modulus of the system to the natural 

 number; and 4343 is the modulus of the common logarithms, when the 4 

 places, next following the characteristic, are taken as whole numbers, instead of 

 decimals, which is meant to be done in the use of the foregoing formula. There- 

 fore—— being very small with respect to b, we shall have, variation of log. b : 



variation of i ( = ■— ) :: 4343 : b very nearly, and thence variation of log. b = 



, vb ^ 4343 , 4343d ,,^, . , , .^. .„„. , ^ 



— 5i^^ ~T~ ~ — ~5iK' Which (puttmg K = 178) = ± 0.452d. Hence 



\og.{b + — ) = log. b + 0.452D; which being substituted in the formula 

 above, will give the difference of height, of the two stations, in English fa- 

 thoms, in a more convenient expression, namely (log. b — log. b ip 0.452d) X 



(1 H ITq~)» where the upper sign, — , is to be used, when the thermometer 



of the barometer is highest at the lower station, and the lower sign, -|-, is to be 

 used, when the said thermometer is lowest at tiae lower station. The first case 

 will be most common ; especially where the difference of height of the two sta- 

 tions is considerable. It should also be observed, that when p, the height of 

 Fahrenheit's thermometer, is less than 40°, then -f ~ , becoming negative 

 or subtractive, must be applied in the calculation accordingly. 



It may perhaps be convenient to repeat here the meaning of the algebraic 

 terms, used in the foregoing formula, that any person may make use of it, 

 without having occasion to recur to the foregoing investigation, b signifies the 

 observed altitude of the barometer at the lower station, and b that at the upper 

 station; log. b and log. b, signify their logarithms taken out of the common 

 tables, by assuming the first 4 figures next following the characteristic as whole 

 numbers, and considering the 3 remaining figures, to the right hand, as deci- 

 mals; D signifies the difference of height of Fahrenheit's thermometer, attached 

 to the barometer at the top and bottom of the hill ; and p signifies the mean of 

 the two heights of Fahrenheit's thermometer, exposed freely for a few minutes 

 to the open air in the shade, at the top and bottom of the hill. 



The formula, for the measure of heights, may also be changed, and adapted 

 to thermometers of particular scales, for the convenience of calculation, as M. 

 de Luc has done ; but these scales will be different from his. The thermometer 

 attached to the barometer, had better be divided with the interval between freez- 

 ing and boiling water, consisting of 81.4 degrees (= 180 X .452) the freezing 

 point may be marked O, and the point of boiling water will be 81.4; for then. 



i 



