the Depression of the Mercury in the Tubes of Barometers. 9 



q being a constant, and Q, Q' &c. functions of t. By differen- 

 tiating, 



d — 

 ' t dQ « dQ! , 



d* d* d< 



+ -^-(3Qm s + 5Q' M 5 + &c.) 



war v 



Substitute the value of — ~ , and equate the coefficients of the 

 two equal quantities, then 



d.Qfi A iV 



&dt t 



and hence, Q = -^ + -^- + &c. 



These coefficients being known, we have the value of the con- 

 stant q, which it is most convenient to arrange according to 

 the'powers of t, viz. 



U , / tt 3 W 5 \ 



S , = T- < Cl8 + We) 



1 Km 



T 1024/ 



It has already been noticed that the depression is the value of 

 — when x and t are both evanescent. But it is obvious that 



the vanishing fractions, — -, -, — have all the same limit : 



wherefore q is the depression. It does not however belong 

 to the surface of the mercury in the tube, but to a surface 

 increased fourteen times in its linear dimensions. To get 

 the real depression we must therefore divide by 2/3. Now 



t = |3r = — l 9 and f = 12*25 I 2 ; and we thus obtain the fol- 

 lowing formula for the real depression, 



U ,/ M 3 M 5 \ 



* "" 49/ \ 192 + 1024/ 



V 2304 T 4096 ) 



The symbol u stands for a known function of s ; but in prac- 

 tice it will be most convenient to have q expressed immediately 

 New Series. Vol. 3. No. 13. Jan. 1828. C in 



