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XLIII. Ohservalions relating to the Depression of Mercury in ■ 

 Glass Tubes ; occasioned by an Article in the last Quarterly 

 Journal of Science. By James Ivory, M.A. F.R.S. 



To Mr. Tilloch. 



Sir, — 1 HAVE to beg the favour of your inserting in your next 

 publication, tlie following observations occasioned by an article 

 in the last Quarterly Journal of Science. Thev relate to the de- 

 pression of mercury in glass tubes, for computing which two tables 

 have been published in the Supplement of the Encyclopcedia 

 Brilannica ; one under the head of Cohesion of Fluids, and 

 the other under that of Fluids. 



It is necessary to begin with saving a few words of the series 

 by which the first of the two tables is constructed. If we neglect 

 all the terms of each of the coefficient-serieses, except the first, as 

 may be done in capillary tubes with very minute bores, the series 

 for computing the depression will become 



s = (M + ^ •(^^•)''+ -£ {i'^y + ^{1^^+ &c. 



Now here, all the coefficients after the first being small, the 

 product Ix, which I consider as the quantity sought, will not be 

 much different from .v ; and it is manifest that, when 5 is consi- 

 derable, a great number of the terms must be taken in, if the ap- 

 proximation is to be carried to many figures, the coefficients de- 

 creasing slowly. In the case of glass and mercury, s is nearly ^ ; 

 and, taking into account the rate of decrease of the coefficients, 

 the total convergency of the scries may be reckoned at ^ ; and the 

 first four or five terms will give an approximation extending to 

 five figures. Accordingly it will be found that, in the table, the 

 depressions for the smallest bores are nearly exact. 



But the case is different when the diameters of the tubes liave 

 a greater magnitude. Then the values of the coefficient-serieses 

 increase greatly, and cannot be computed with any degree of cer- 

 tainty from two or three of their first terms. In reality all these 

 serieses, excepting the first which is very regular and convergent, 



diverge in their first terms instead of converging, when ^ is 



nearly equal to, or greater than, unit. The divergency indeed 

 goes on to a certain point only, after which the serieses will con- 

 verge, and they will ultimately converge very rapidly as in the 

 first one. In this manner of computing therefore, when the tubes 

 have large bores, the degree of exactness in the result will de- 

 pend entirely on the three first terms of the value of the sine of 

 depression ; since these terms are the only ones in which the co- 

 L 1 2 efficients 



