82 CORRECTION OF THE HEIGIIT OF THE BAROMETER. 



the highest powers of b form this series, — , — , 



b v 2 3 3X4* 



— — ■, which may be continued at pleabure, and which 



3.4.5 * 



must obviously express the versed sine of a circular arc, 

 since, when b becomes infinite, the curve coincides with a 

 circle, and the highest powers of b must in this case be in- 

 finitely greater than all the rest : the coefficients of the 



powers of b next in order form this series, , tt— r. 



r 4.4 6.6' 



10 14 10 14 18 __10 14 18 22 



sTi ' T' hk io * 2 ' 3 ' 12T12 ' "2" ' T ' T' an 



those of the other orders of terms seem to follow a law 

 nearly similar, but which I have not fully ascertained. The 

 coefficients of the terms including the lowest powers of b 

 are however of more consequence; the progression of the 

 coefficients of b is sufficiently obvious : those of b 3 have 

 their denominators increased according to the same law, 

 and the ratio of the numerators approximates to 8, and dif- 

 fers so little from it, that this number may be employed as 

 a multiplier without sensible errour: and in a similar man- 

 ner if the denominators of the coefficients of b 5 are made 

 to increase in the same ratio, the numerators approach 

 nearer and nearer to the ratio of 16 to 1, which is probably 

 their ultimate proportion: and in some instances the conti- 

 nuation of the progressions on these principles is required 

 for a sufficiently accurate determination of the quantities 

 concerned. 

 Arrangement For calculating the depression or elevation of a fluid in a 

 of the series. ^ u ^ e Q f a gj ven diameter, it is convenient to arrange the 

 series according to the powers of b; so that the whole as- 

 sumes this form, n rz 



(2 m x 



