the Measurement of the Specific Gravity of Liquids. 115 



when the distances ab, ao and the specific gravity of the liquid 

 are given ; the other part might be calculated if the equation of 

 the curved siu-face acbd were known. 



If y denote the height e^ of any point in the curved surface 

 above the level of the liquid, p and p' the radii of greatest and 

 least curvature of the surface at that point, and 7n a constant, 

 then generally 



y=4l+l^ ^^^ 



an equation first evolved by Young*, and afterwards confirmed 

 by the researches of Laplace t, Poisson J, and Gauss §. Young 

 proceeded on the hypothesis that there existed in the surface of a 

 liquid a sort of tension, which was the same at every point, — an 

 hypothesis which agrees entirely with the later researches above 

 alluded to, though the latter partly rest on other suppositions as 

 well. 



If, now, T denote the tension in a strip of the surface whose 

 breadth is unity, and k the weight of a unit of volume of the liquid, 

 the constant in the above expression is, according to Hagen ||, 



T 

 «^=■| (2) 



At the highest point b of the curve bed (whose ordinate bj) we 

 will indicate by H), p' is equal to the radius of the cylinder itself, 

 and p to the radius of greatest curvature of the curve bed, which 

 by revolution round the axis of the cylinder, describes the capil- 

 lary surface of double curvature. And it is easy to see that 

 these two radii lie in the same straight line, and that the one is 

 positive and the other negative. In this case, therefore, equa- 

 tion (1) takes the form 



'y^^^i-p-}) ^^^ 



Although it is not difficult from expression (2), given above, 

 to find the diff'erential equation of the curved surface, the inte- 

 gration of that equation is involved in great and probably insu- 

 perable difficulty. 



I have therefore adopted the same method as Hagen ^, whereby, 

 although the equation of the surface is not found, m receives a 



* "An Essay on the Cohesion of Fluitls," Phil. Trans. 1805. 



t " On Capillary Attraction," and " Sui)plement to the Theory of Ca- 

 pillarity," M^vanique Celeste, vol. iv. (1805). 



X New Theory of Capillary Attraction (1831). 



§ " Principia {^cneralia theorise iigunc fluidonira in statu aiquiiihrii," 

 Comin..Hoe. Scieni. Gilttinfj. vol. vii. (182!)). 



II Po<;gendortt''s Annalen, vol. Ixvii. p. 1. 

 If Ibid. vol. l.wii. pj). 27 and 28. 



12 



