MEASUREMENT OF HEIGHTS. 539 



but we must not hence conclude Dalton's assumption as not re- 

 concilable, under all circumstances, with the existence in equi- 

 librium of an atmosphere of aqueous vapour of which the den- 

 sity is always a jDositive quantity. The decrease of t does not 

 go on indefinitely, but only as far as the value which it pos- 

 sesses at the limit of the atmosphere ; the formula (9.), which 

 expresses the condition, is merely an interpolation formula, and 

 has no justification beyond its more or less satisfactory accord- 

 ance with Dalton's experiments made between ^ = 0, and t 

 = 100°. 



If we take the logarithm of the two quantities, between which 

 the conditions apply, it follows that 



io« = (^ . ^ - « + c (t + o) (t - 0; 



and we also know that «" 1, so that log « must not be positive. 



Hence it follows that the conditions may be fulfilled, or that 

 the atmosphere of aqueous vapour is possible ; also that the 

 value of a. {< 1), which determines its density at the elevation A, 

 remains arbitrary, if 



f ^>a-c(r+0 .... (19.) 



which must be the case up to the limit of the atmosphere ; fur- 

 ther, that in the opposite case, if even at the height A, 



-|'- ^<a-c(T+0. . . . (20.) 



the existence of an atmosphere of aqueous vapour in equilibrium 

 is possible ; but its density, at the elevation li, is limited by the 

 condition that a. must be less than 



1o(t¥-'^ + ^('-+'0^^-'^- • • (21-) 



for the value of t at the limit of the atmosphere. In a parti- 

 cular case of the decrease of temperature, the atmosphere of 

 aqueous vapour may be at all elevations as dense as the tempe- 

 rature permits ; this case requires that 



dp^_ d, dX __ d 'p t 



~p,~ ~ ^' r +T7 " "^ ' 



or, under the supposition of the customary expression for <f i, 

 that 



