520 BESSEL ON BAROMETRICAL 



elevations x and w -\- dx has for every unit of its surface the 

 mass % .dx; therefore it exerts on this unit the pressure 



(^) ' (^)' ^^' 



in -nhich (_5i') is gravity at that part of the earth's surface ■<;^•h^ch 

 is perpendicularly beneath the point to which x and 8 belong, 

 expressed in terms of gravity in the latitude of 45°. But in 

 order that the diminution of pressure, caused by taking away 

 this stratum from those above x, may be obtained in terms of 

 the measui'e applied to f, the above expression must be divided 

 by that measure, which then gives 



or, if we prefer the use of the toise to that of the Paris line, 



^ 336-905 \a + x) ^ ' 



If we eliminate I by combining the two equations, w^e obtain 



^ [g] 864 . D / « y dx^ 



'p ~ 336-905 \a + x) E ' 



By the integral of this equation the values of p at two different 

 elevations above the surface of the earth, x = h, and x = A', be- 

 come comparable with each other ; or, if we denote them by P 

 and P', and employ Briggs's logarithms, of which the modulus 

 h ft,, 



F _ ig) S64 . D . !«. rh' r a X dx 



loa- -= - (^) S64 . D . jt. r^' { a \ 

 ^ P 336-905 J h \a + x) 



= 9397-74 = l=i9) I' 



E 



or if we write 



336-905 

 8G4 . D . ;u. 

 then 



lor 



P - _ 1 /•'7_JL-V ^ (.) 



P /' Jh \a-\- x) E • • • ^""'^ 



The integration, which still remains to be performed, requires 

 that we know the relation between x and E, or the law according 

 to which the observed heights of the thermometer t and t', cor- 

 responding to the temperature of the air at the two heights, pass 

 into each other. We do not know this law in every case, and 

 we have, therefoi'e, no ground for assuming the change of tem- 

 perature to be otherwise than proportioned to the change of ele- 



