134 SMITHSONIAN MISCELLANEOUS COLLECTIONS VOL. 51 



(6) The vapor of water is transformed partially into water. 

 Approximately we have 



a (273 + r) 

 p V = (p- f)v' + f v" x = + x f v" 



By the substitution of this equation in equation (11) and by the 

 aid of the formulas of case No. 3. we find 



(12) 



.(13) 



= (c -f £ J) dx + d (x I) + A (I + £) d h 



= c + £ c') (r -t ) + x I- x l + A(l + £) h 



(7) Congelation at o°. 

 In this case we find 



= Idx -Ldz + A(z + $)dh } 



= (/ + L) (x - x ) - Ly + .4(i + t)h j 



(8) The vapor of water is partially transformed into ice. 

 We find 



= (c + £ c") (r - t ) + x(l + L) - x (l + L) +A(i + £)* (14) 



All these formulas tliat we have developed, pertain to the case 

 in which the air experiences transformations without gaining or 

 losing heat. Let us suppose that the air receives some heat and 

 that the heat absorbed is proportional to the variation of the tem- 

 perature. When the quantity of absorbed heat is small, the tem- 

 perature decreases during the expansion and in place of equation 

 (1) we write 



- bdz = d U + A pdV (15) 



Here b expresses a constant which depends on exterior circumstan- 

 ces. Applying this equation to the dry air we find the law given by 

 formula (2) and m is given by the equation 



m = 3.441 ( 1 +- ) 06) 



We see that we can easily apply equation (15) to other cases in 

 which the vapor is condensed, but we refrain from the development 

 of the formulas because there are no observations wherewith to 



