8 Mr. C. Chree on the 



That Beetz's formula is incorrect has been clearly pointed 

 out by Weber *. He looks on the liquid as bounded by two 

 infinite cylinders and supposes the thermometer, the sur- 

 rounding mercury, the inner glass vessel, and the inner 

 surface of the liquid to be at one temperature. He thence 

 deduces for the temperature-difference the equation 



k „ 



where t is the time, k the conductivity of the liquid, p the 

 density and c the specific heat of mercury, w r hile A is a con- 

 stant. The summation extends to all values of m given by 

 the equation 



JojmrijY^mrz) — J (mr 2 )Y (mr 1 ) _ 2pc m 

 mri $ % imri) Y (mr 2 ) — J (mr 2 ) Y^rnr^ p x Ci 



where r x , r 2 are the radii of the surfaces bounding the liquid, 

 whose density is pi and specific heat c l5 and J, Y stand as 

 usual for the two solutions of the Bessel's equation of the 

 degree indicated by the suffix. Supposing r 2 —r 1 small com- 

 pared to r ly it is easily proved that the least root of the above 

 equation is much smaller than the next. This explains why, 

 except at the commencement of the heating or cooling, the 



retention of but a single term, viz. 7—Ke~P c , gives a fairly 

 satisfactory result. 



Weber contents himself with proving the falsity of Beetz's 

 equation. It is not, however, difficult to obtain an approxi- 

 mate value for the least value of m, and thus to deduce from 

 Beetz's observations fairly satisfactory values for the conduc- 

 tivity. Thus, supposing m(?' 2 — ^i) small, we may put 



J {mr 2 ) = Joimri) + m (^-nW ( mr \) > 

 where T , d T 



mJ ° dr J °> 

 and use a similar result for Y (mr 2 ). 

 Also J ' = — Jj, and Y(/=— Y x ; 



thus the equation reduces to 



whence _ nk_ t 



T = r e pi c i , 



where n is constant for the apparatus. We should thus get 

 k a p! Ci C ; i. e. we have only to multiply Beetz's numbers by 

 the specific heat of unit volume of the liquid considered to 



* Wied. Ann. Bd. x. pp. 480-490. 



