266 Prof. Potter on some properties of the Air Thermometer. 

 Multiplying each 'side by 2 . ~ and integrating, we have 



dfi pV }V a + b-J"*' 



' doc 



or putting v = the velocity of the surface at P= -j-, then 



and the velocity is at A when #=0 ; 



^{a + ia + tyhg^l+C. 

 Subtracting, we have the corrected integral 



^{(« + J)log t p^)-(*-,)}. 



Prom the correction of the integral, this will always give the 

 velocity v=*0 when x = a; and there will be another value of a? 

 when v = 0, as A'O, to be determined by the equation 



o=ag 



, fa-\-b—x\ a—x 



losl —b—)=j+b> 



Oe 



of which the value of the roots can be found by trial and suc- 

 cessive approximations in terms of given values of a and h. 

 These give as follows the values of x when v=0. 

 When 



a is small compared with b } the roots area: = -\-a and x = — a; 

 a = bj „ x= a „ x = — l*513tf; 



a — 2b, „ x— a „ x—— 1*8560; 



a = 3bj „ x= a „ x = — 2'116d; 



we see that if there were no adhesion of the liquid to the tube, 

 and no heat or cold developed, there would be unsymmetrical 

 oscillations of the surface at P about the point 0, similar to what 

 we find in certain other dynamical problems. 



Next, taking the case of the vertical barometer gauges or ma- 

 nometers, as used in all the experiments, together with the adhe- 

 sion as proportional to the adhering surfaces of the liquid and 

 the tube, we have the following proposition : — 



Prop. To determine the motion of the liquid in the vertical 

 gauge of an air thermometer, taking into account the adhesion of 

 the liquid to the tube } when it arises from sudden changes of den* 

 sitij of the contained air. 



