VOL. XXXVIII.] PHILOSOPHICAL TKANSACTIONS. 50) 



from the vessel v, that is, the surface p must subside so far, that a cubic inch 

 of water, if that be the fluid made use of, shall enter the vessel v ; by which 

 means the barometer, with the parts annexed, will be heavier by a cubic inch 

 of the fluid. 



Now this additional weight of a cubic inch of fluid, will make the whole 

 barometer subside, according to the laws of hydrostatics, till a cubic inch of 

 the rod hs, immediately above the surface at w, shall come under it; but the 

 length of such a magnitude of hs will exceed the length of an equal magnitude 

 of quicksilver in the larger tube x, as much as the square of the diameter at x, 

 exceeds the square of the diameter at h, the lengths of equal cylinders being 

 reciprocal to their bases. That is, the perpendicular descent of the compound 

 barometer will be to v, the perpendicular ascent of the mercury in the common 

 barometer, as (/ to 1, supposing this the ratio of the bases; and consequently 

 will be equal to dv. 



But by this descent, the distance pw, between the surface of the stagnant 

 quicksilver and the top of the fluid, will be augmented by a column, whose 

 height is dv, the descent of the compound barometer; consequently the weight 

 of the whole column of the fluid, pressing on the lower surface of the quick- 

 silver, to which the height x is partly owing, will be increased by a column of 

 that length; and this increase would produce a second ascent of the mercury 

 at X, equal to itself, namely dv, were the fluid as heavy as quicksilver; but 

 since it is supposed to be lighter in the ratio of i to J , the ascent of the mer- 

 cury on this account will be only — . 



But now, as in the former case, when the ascent of the mercury was v, the 

 descent of the compound barometer was shown to be dv, so here the ascent of 

 the mercury being — , the descent of the compound barometer will be — , the 



next descent -^, the next -4, and so on to infinity. Therefore the whole de- 



scent of the compound barometer, is to the ascent of the mercury in the com- 



d^ . d' . d* . „ . . - *_ I 



■d 

 because the terms of the series being in geometrical progression, the sum of 



them all is — ^,. Hence we have n = — —,, and ns = ds -\- dn; that is, 1 : d :: 

 s — d s — d 



n -\- s: ns :: '^-^ : 1 and 1 : ^d, that is, the diameter of st to the diameter of 



Hi, as ^/'^-^ to 1. a. E. D. 



Example 1. — Putting s = 14, and n= ], the variations in each barometer 

 will be equal, by taking the diameter of st to the diameter of hi, as v/— : 1, 

 that is, as 30 to 29 nearly. 



