682 PLATEAU ON THE PHENOMENA OF A FREE LIQUID MASS 



uniformity so much the less as the charge is greater ; whence we 

 may infer, that for sufficiently great charges, the length of the 

 continuous part of the real vein must still exactly follow this 

 law. We shall, moreover, demonstrate this in a rigorous manner. 



73. Let us then take the real case, i. e. let us consider a vein 

 submitted to the action of gravity, in which consequently the 

 movement of transference is accelerated. Then the velocity 

 possessed, after any time t, by a horizontal section of the liquid 

 conveyed by the movement of transference, will have for its value 

 \^2ffh + f/t, the first term representing the portion of the velo- 

 city due to the charge, the second the portion due to the action 

 of gravity upon the vein, and t being reckoned from the moment 

 at which the liquid section passes the contracted section. It 

 must be borne in mind, that in virtue of the acceleration of the 

 velocity, the vein, if it did not become divided, would continue 

 to become indefinitely thinner from above downwards (§ 69). 



This admitted, let us imagine that another vein of the same 

 hquid, placed under the imaginary condition of the preceding- 

 paragraph, flows off with the same charge from another orifice of 

 the same diameter, in the same time as the true vein in question. 

 Let 6 denote the time occupied by this second vein in traversing 

 the distance which we have denoted by D, i. e. that which is 

 comprised between the instant at which the liquid section that 

 constitutes the neck of a constriction passes to the contracted 

 section, and the instant of the rupture of the line into which 

 this constriction becomes transformed. In the expression of 

 the velocity relative to the first vein, let t = 9, which gives for 

 this velocity, after the time 6, the value \/2gh + g6', in other 

 words, let us consider the velocity of a liquid section belonging 

 to the true vein, after the time necessary for a section belonging 

 to the imaginary vein to have traversed the distance D. Accord- 

 ing to what we have seen in the preceding section, if the orifice 

 remains the same, this time is constant whatever the charge 

 may be ; so that in the above expression the term gd remains 

 invariable when h is made to vary. Hence whatever may be the 

 value of 6, we may suppose the charge h to be sufficiently large 

 for the term V2gh to be very great in proportion to the term^^, 

 and that the latter consequently may be neglected without any 

 sensible error. In the case of a value of h which will realize this 

 condition, and a fortiori in the case of all still greater values, 

 the velocity of a section of the true vein during the time 6 may 



