Trowbridge.] O-jO [Nov. 3, 



the increase in volume for an increase in temperature of 10 Fab., so tlial if 

 Vo be any volume at tbe temperature of 32^^, it will become Vq (1 + -< to) 

 = V, at the temperature to above 32°; and at the temperature t above 32° 

 the volume will be 



VJ(l + xt):^V.(^-,p^J=V (1). 



Since the pressure is equal to the weight, we have 



^p=/>^G-TiioV' • ^'^- 



This equation applies for the hight z. We have 



Vi = * rr [(r + zj" — r*], and dVj = 4 - (r + z)Mz (3). 



gr* tHo 



z= (FT^. (4)> ^ = (r+^^ (5)- 



Mariotte's Law gives 



p=|_/,, dp= J^dp (6). 



If we substitute these values in (2), and make the second member nega- 

 tive, since f, decreases as z increases, we shall have 



— .„ — _ 4 _ er^ ^ ^ fl + -i-l^l dz (7). 



p d/> - *-g^P(i+xto) L ^(r + z)d 



lie integral of this gives 



Log./> + C=-4^gr^ P(r+Tt7) L"-T+z^J 



rA;.to 

 When z = 0, « = A, and C + log. A = 4 ?: gr^ -_-——, and 



P(l + /to) 



Log. .L =-4.gr-. pj^-^^ L^+ ;-Tt,- ^.or 



^--^— ^--po^t^-^] («^ 



Now make 



P R ^''^' 



then we have 



«'"='■ ^=-rTlf;[^ + TfV] <««■ 



Let us take the surface of the earth as the unit of surface, the radius of 

 the earth as a linear unit, and the force of gravity at the earth's surface as 

 the unit of gravity. 



' 2. Let D' be the density of mercury at the temperature of 32^ and D" 

 its density at to above 32° ; then, if /' be the coefficient of expansion for 

 a volume of mercury, 



D' = (1 + /' to) D" (11). 



Let h' be the hight of a column of mercury on the unit of surface, and 

 at the temperature of 320, that will just balance a column of atmosphere 



