476 



OnCapil- width of 



HYDRODYNAMICS. 



sion of 

 Fluids. 



I Aurc - plane, or what is the same thing, to the 



nun and " i nter ' or circumference of the tube. Calling c, there- 

 tlie Cohe- fore, the circumference of the tube, we shall have 

 Q= {c; 5 being a constant quantity, representing the 

 intensity of the attraction of the tube AB upon the 

 fluid, in the case where the attractions of different bo- 

 dies are expressed by the same function of the distance. 

 In every case, however, 5 expresses a quantity de- 

 pendent on the attraction of the matter of the tube, 

 and independent of its figure and magnitude. In like 

 manner we shall have Q'= g'c; 5' expressing the same 

 thing with regard to the attraction of the fluid for it- 

 self, that 5 expressed with regard to the attraction of 

 the tube for the fluid. By substituting these values 

 of Q> Q'> in the preceding equation, we have 



If we now substitute, in this general formula, the 

 value of c in terms of the radius if it is a capillary 

 tube, or in terms of the sides if the section is a rect- 

 angle, and the value of V in terms of the radius and al- 

 titude of the fluid column, we shall obtain an equation 

 by which the heights of ascent may be calculated for 

 tubes of all diameters, after the height, belonging to any 

 given diameter, has been ascertained by direct experi- 

 ment. 



Application In the case of a cylindrical tube, let a represent the 

 of the for- rat ; o o f the circumference to the diameter, h the height 

 1 c) ~ of the fluid column reckoned from the lower point of 

 the meniscus, q the mean height to which the fluid 

 rises, or the height at which the fluid would stand if 

 the meniscus were to fall down and assume a level sur- 

 face, then we have vi 5 for the solid contents of a cy- 

 linder of the same height and radius as the meniscus, 

 and as the meniscus, added to the solid contents of the 

 hemisphere of the same radius, must be equal to * r 1 , we 



have ?rr 3 , or , for thersolid contents of the 



o o 



meniscus. But since ^ = ?rr 2 X -5-, it follows that 

 o o 



tr r 3 



the meniscus - - is equal to a cylinder whose base 



tubes. 



height of the fluid in his 2d tube by means of this con- On Capil- 



1. 90381 lary Atirac- 



stant quantity, we have r = - = 0.951905, and tion and 



the Cohc- 



2 p' i 15 1311 sion ol 



2 -yijXy =q =^=15.8956, from which, fl ^- 



if we subtract one sixth of the diameter, or 0.3173 

 we have 15.57S3 for the altitude h of the lower point 

 of the concavity of the meniscus, which differs only 

 0.0078 from 15.861 the observed altitude. 



If we apply the same formula to Gay Lussac's ex- 

 periments on alcohol, we shall find the constant quan- 



Application 

 of the for- 



& p p mula toGay 



tll y 2 ~ TTT^ = 6.0825 as deduced from the 1st ex- Lussac's ex. 



penmen ts 

 on alcohol. 



periment, and h = 6.0725, which differs only 0;0100 

 from 6.08397, the altitude observed. 



From these comparisons, it is obvious, that the mean 

 altitudes, or the values of q, are very nearly reciprocal- 

 ly propoitional to the diameters of the tubes; for, in 

 the experiments on water, the value of q deduced from 

 this ratio is 15.895, which differs little from 15.9034, 

 the value found from experiment; and that in accurate 

 experiments, the correction made by the addition of the 

 sixth part of the diameter of the tube is indispensibly 

 requisite. 



If the section of the pipe in which the fluid ascends 

 is a rectangle, whose greater side is , and its lesser side 

 d, then the base of the elevated column will be = a d,. 

 and its perimeter c= 2 a + 2 d. Hence, the value of 



the meniscus will be ~~ - ^L ( i _ M 



that is q h + f 1 _ ~\ Hence, if in the ge- 

 neral equation No. 1. we substitute for c its equal 

 2 a-f-2 d, and for V its equal adq, we have 



Application 

 of the for- 

 mula to 

 rectangular 

 capillary 

 spaces. 



=2 'X 2a + 2 

 and dividing by a and by g D, we have 



, and 

 a 



is v r*, and altitude . Hence, we have 



er what is the same thing, the mean altitude q in a cy- 

 linder is always equal to the altitude h of the lower 

 point of the concavity of the meniscus increased by one 

 third of the radius, or one sixth of the diameter of the 

 capillary tube. Now, since the contour c of the tube 

 2 sr g, and since the volume V of water raised is 

 equal to q X -x r", we have, by substituting these values 

 in the general formula, 



gD g5 rr z =:2*r(2 s . ? '), (No. 1.) 

 and dividing by tc r and g D, we have, 



In applying this formula to the elevation of water 

 between two glass plates, the side a is very great com- 

 pared with d, and therefore the quantity being al- 

 most insensible, may be safely neglected. Hence the for. 

 mula becomes 



Application 

 of the for- 

 mula to 

 Gay Lus- 

 sac's expe- 

 riments 

 on water. 



t ~" " TTV ^> I * 



gD d 



By comparing this formula with the formula No. 2. 

 it is obvious, that water will rise to the same height 

 between plates of glass as in a tube, provided the dis- 

 tance d between the two plates of glass is equal to r, 

 or half the diameter of the tube. This result was ob- 

 tained by Newton, and has been confirmed by the ex- 

 In applying this formula to Gay Lussac's experi- periments o f succeeding writers. 



raents, we have the constant quantity 2 2g - g =: As tne constan t quantity 2 ^~ is the same as al- 



r-X-^- ( No -*-) 





rq = 647205 X 23,1634. + 0,215735= 15,1311 for 

 Gay Lussac's 1st experiment. In order to find the 



^ 



ready found for capillary tubes, we may take its value, G Lus 

 viz. 15,1311, and substitute it in the preceding equa- sac > s experi- 

 tion, we then have ments. 



3 



