Now, the fluid in the lower part of the round tube AB is attracted, 1. By itself; but as the reciprocal attractions of a body do not communicate to it any motion if it is solid, we may, without disturbing the equilibrium, conceive the fluid in AB frozen. 2. The fluid in the lower part of AB is attracted by the interior fluid of the tube BE, but as the latter is attracted upwards by the same force, these two ac tions may be neglected as balancing each other. 3. The fluid in the lower part of BE, is attracted by the fluid which surrounds the ideal tube BE, and the result of this attrac tion is a vertical force acting downwards, which we may call Q', the contrary sip being applied, as the force is here opposite to the other force Q. As it is highly proba ble that the attractive forces exercised by the glass and the water vary according to the same function of the distance, so as to differ only in their intensities, we may employ the constant coefficients c' as measures of their intensity, so that the forces Q and Q' will be proportional to g ; for the interior surface of the fluid which surrounds the tube BE, is the same as the interior surface of the tube AB. Consequently, the two masses, viz. the glass in AB, and the fluid mound BE, differ only in their thickness ; but as the attraction of both these masses is insensible at sen sible distances, the difference of their thicknesses, provided their thicknesses are sensible, will produce no difference in the attractions. 4. The fluid in the tube AB is also acted upon by another force, namely, by the sides of the tube All in which it is inclosed. if we conceive the column hB di vided into an infinite number of elementary vertical co lumns, and if at the upper extremity of one of these co lumns we draw a horizontal plane, the portion of the tube comprehended between the plane and the level surface BC of the fluid, will not produce any vertical force upon the column ; consequently, the only native vertical force is that which is produced by the ring of the tube immediately above the horizontal plane. Now, the vertical attraction of this part of the tube upon BE, will be equal to that of the entire tube upon the column BE which is equal in diameter, and similarly placed. This new force will therefore be repre sented by + Q. In combining these different forces, it is manifest that the fluid column BF is attracted upwards by the two forces Q, Q, and downwards by the force Q'; consequently, the force with which it is raised up wards will be 2 Q Q'. If we represent by V the volume of the column BE, D its density, and g the force of gravity, then g DV will represent the weight of the elevated co lumn ; but as this weight is in equilibrio with the forces by which it is elevated, we have the following equation : g DV= 2QQ.
If the force 2Q is less than then V will he nega tive, and the fluid will sink in the tube ; but as long as 2Q is greater than Q', V will be positive, and the fluid will rise above its natural level ; as was long before shown by M. Clairaut.
Since the attractive forces, both of the glass and the fluid, are insensible at sensible distances, the surface of the tube AB will act sensibly only on the column of fluid immediately in contact with it. We may therefore ne glect the consideration of the curvature, and consider the inner surface as developed upon a plane. The force Q will therefore be proportional to the width of this plane, or, what is the same thing, to the interior circumference of the tube. Calling c, therefore, the circumference or the
tube, we shall have Q=cc; c being a constant quantity, representing the intensity of the attraction of the tube AB upon the fluid, in the case where the attractions of differ ent bodies are expressed by the same function of the dis tance. In every case, however, c expresses a quantity de pendent on the attraction of the matter of the tube, and in dependent of its figure and magnitude. In like manner we shall have c ; c' expressing the same thing with regard to the attraction of the fluid fur itself, that e ex pressed with regard to the attraction of the tube for the fluid. By substituting these values of Q, Q', in the pre cc ling equation, we have g DV = c (2 ce).
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 rectangle, and the value of V in terms of the radius and altitude 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 as certained by direct experiment.
In the case of the c) lindrical tube, let ,r represent the ratio of the circumference to the diameter, /: the height 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 surface, then we have Trr3 for the solid contents of a cylinder of the same heignt and radius as the meniscus ; and as the meniscus, added to the solid contents of the hemisphere of the same radius, 2 s r3 must be equal to a we have or 7,,for mr 3 the solid contents of the meniscus. But since 1' it follows that the meniscus 3 is equal to a cy - linder whose base is ,r and altitude 3 . Hence, we have q = h ± 3 ; or, what is the same thing, the mean altitude q in a cylinder 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 = 2a' e, and since the volume V of water raised is equal to q x we have, by substituting these values in the general formula, g D = 27r r e g'), (No. 1.) and dividing by vr r and g D, we have, 2 e e _e r q = 2 and q = 2 2X . (No. 2.) D In applying this formula to Gay Lussac's experiments, we have the constant quantity 2 2 r q = 647205 g X 23,1634 + 0,215735 = 15,1311 for Gay Lussac's 1st ex periment. In order to find the height of the fluid in his 2d tube by means of this constant quantity, we have r = 1.90381 2 g e' 1 15,1311 ---.= 0.951905, and 2 x = q= 2 gD 3 0.951905 = 15.8956, from which, if we subtract one sixth of the diameter, or 0.3173, we have 15.5783 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 experi ments on alcohol, we shall find the constant quantity e = 6.0825 as deduced from the 1st experiment, g and h = 6.0725, which differs only 0.0100 from 6.08397, the altitude observed.