268 Oil the Depression of Mircilry in Glass Tubes. 



efficients of the unknown quantity can be computed with ari^ -m 

 tolerable precision. 



If the exactness of the result would be affected in the fourtlij 

 and fifth deciuial places by omitting entirely all the terms in the] 

 value of the sine except tlie three first, it is to be feared that a 

 like inaccuracy will occur although some of the omitted terms be 

 taken in, but with inexact coefficients, greatly short of the real 

 Values, not amounting perhaps to the 10th, or the lOOdth, of 

 even the lOOOdth, part of the truth. 



Such appears to be the objection to this mode of computation; 

 and the cbefficient-serieses, after the first and second, are so com. 

 plicated that the disadvantage does not .Appear to admit of a re- 

 medy. Still however the method is possessed of considerable ac- 

 curacy, which it owes to the circumstance, noticed by the author, 

 that the first term alone brings out the truth within a ^ih of 

 the whole. 



The rulies by which the other table is constructed are ihvesti- 

 gated on this principle; that the quantity denoted by ^ is very 

 nearly equal to unit in the case of small capillary tubes, and even 

 in the largest bores it decreases oidy to a certain limit wliich is 

 greater than ^i-. A formula is therefore sought for determining 

 /' when it differs sensibly from the limit ; and in all other cases, 

 the same quantity is supposed to coincide with the limit. Ths 

 rules in the article Fluids in the Supplement to the EncyclopcB- 

 dia will determine the depression to five places of figures, and 

 are therefore more than sufficient for any practical purpose. But 

 it is by no means supposed that it is impossible to find other and 

 better rules for the same purpose ; as I shall presently show by 

 giving another formula, which has the advantage of determining 

 the depression directly from the given quantities without the so- 

 lution of an equation. I put q for the depression sought ; If 

 for the diameter of the tube; z, for the sine of depression 



t= '735; t = (^yY; and 



A=l + -^.4- + — .^ + —^ . - + &c. ; then 



5~ 49.;.>. T*A3 • ( 12 "^ 'W "^ 1440 "^ 1200^ 



4 * A.S • i 32 "^ 384 "" 1920 "*" 161280* 



"4 ' XT ' I 64 "^ 1920 "'' 32^56 ' 



4 * a9 • i 768 3072 ' 2580480' ^' 



This formula will extend to all tubes not much exceeding six- 

 tenths of an inch ; and by means of it the results in the tablfe 



niav 



