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THE CIVIL ENGINEER AND ARCHITECTS JOURNAL. 



123 



tide on the subject ; but an explanation of them so clear and so ob- 

 vious has occurred to me ; and even your own article contains such 

 ample proof of the truth of my assertions on the main point at issue 

 between us, namely, the velocity of water in a vertical pipe, that even 

 at the risk of being charged with presumption, and under all the dis- 

 advantages of endeavouring to maintain a discussion with one supe- 

 rior to me in scientific acquirements, I am tempted to offer you a few 

 more remarks. 



" I think it will be well to treat first ' the velocity of the water in 

 the pipe,' that being the chief point of difference, and also because, 

 succeeding in proving this, I think it will appear as a necessary 

 consequence, that the uniform flow of water in the pipe is quite in- 

 dependent of the cohesion existing between ' one particle and that 

 immediately above it.' 



" You state, and evidently quite correctly, that, supposing a vertical 

 pipe 16 feet in length full of water to have the supporting base re- 

 moved ' it would be emptied by the fall of water in one second, the 

 final velocity of the water on issuing from the pipe, would be 32 feet 

 per second, and the mean of the initial and final velocities would be 

 16 feet per second.' This is evidently quite correct. The whole 

 column of water falls like a block of marble ; the upper and lower 

 parts are equally subject to the action of gravity, and every particle 

 of water descending equal spaces in equal times, the whole column 

 will fall at once, and this even without supposing that ' each particle 

 coheres to the particle immediately above it,' and quite indepen- 

 dently of cohesion, except, of course, what I may term, lateral cohe- 

 sion, or that existing between the particles of the same horizontal 

 section. The momentum of such a column is, therefore, 16 feet mul- 

 tiplied by the weight of the water; but you also state that, if the 

 pipe had not been permitted to empty itself, but had been maintained 

 constantly full, the velocity of the column of water would be 16 feet, 

 and the momentum consequently the same as in the former case. 

 Would not this conclusion be sufficient to convince you that there is 

 an error in your reasoning? 



" In the first case, the momentum, according to your reasoning, is 

 the same as in the second, when the pipe is maintained constantly full. 

 Surely this must be wrong. The mistake appears to me to arise from 

 your not taking into account that, the pipe being supposed constantly 

 full, the water must enter as fast as it issues, whether this entering 

 velocity is imparted to the water by the. height of the reservoir or 

 other force. If the height of the reservoir or other force, which 

 supplies the pipe with water, is not sufficient to impart to the 

 entering water a velocity equal to that due to the whole height 

 of the pipe, then, I maintain that the height of the water in 

 the pipe will diminish proportionally, so as to regulate the issuing 

 velocity according to the supply. I shall endeavour to explain this. 

 No doubt, still supposing the pipe to be constantly full, and, as before, 

 to be IS feet in length, in the first second, the mean velocity will be 

 only 16 feet per second, but for the remainder of the time, the ve- 

 locity will be 32 feet per second, as will be apparent by carefully con- 

 sidering the following explanation. 



" Let us first examine the velocity of the water at different points 

 of the pipe in the first case; when the pipe being supposed full, and 

 the supporting base removed, the water is allowed to fall freely, ac- 

 cording to the law of gravitation: that the velocity at any point 

 varies as the square root of its distance from the top of the pipe. 

 At one fourth of the height, the velocity of the water, therefore, 

 ■will be 16 feet per second ; and at a foot from the top the velocity 

 will be four feet per second; therefore, to keep a pipe one foot long 

 constantly full, the water must flow in at the rate of four feet per 

 second ; and to maintain one four feet long full, the water must enter 

 at 16 feet per second ; if the reservoir does not supply water at this 

 velocity, the pipe will not be perfectly full; and by analogy and fol- 

 lowing the same train of reasoning, we deduce that the pipe being 

 16 feet in length, the velocity will be 32 feet per second, and the 

 water must enter the pipe at that rate. Suppose, for instance, the 

 area of the pipe to be one square foot, if the reservoir supplies 32 

 cubic feet of water per second, then the pipe of 16 feet in length will 

 be maintained constantly full; if the reservoir can only supply 16 

 cubic feet, then the pipe 16 feet in length will be maintained full" only 

 to oue fourth of its height or four feet from the bottom ; or let us 

 suppose a glass tube 16 feet in length to be maintained full by pouring 

 water from a jug, and let some colouring liquid, of the same specific 

 gravity as water, be placed at the top, it will be found that as the 

 colouring liquid descends, the velocity of the water poured in must 

 be gradually increased until the colouring liquid has reached the 

 bottom of the tube, when the velocity will be found to be 32 feet per 

 second ; during this second, therefore, the mean velocity will be 16 

 feet, and during the remainder of the time the velocity of the water 

 to maintain the tube full must be 32 feet per second. If you do not 



pour in the water at this velocity, you will find, as I before stated 

 that the tube will not be maintained full, but that the surface of the 

 water will fall until the height of the filled part corresponds with the 

 velocity of the supply. 



" I trust I have proved even to your satisfaction that my reasoning 

 on this point is correct. I have shown, from your own admission, 

 viz., ' that the final velocity of the issuing water from the pipe in the 

 first class, would be 32 feet per second, and the mean between the final 

 and initial velocities 16 feet per second ;' that to maintain a pipe 16 

 feet in length constantly full, requires the water to enter at the velo- 

 city of 32 feet per second, and, rice rend, if the idpe be maintained 

 constantly full, that the issuing velocity will be 32 feet per second. 

 The above law is true of any body in vacuo; it is true of shot in 

 vacuo ; it would be true of shillings, in pleno, falling through a tube 

 without friction, and fitting accurately the sides of the tube, so as to 

 prevent the air from entering at the sides ; thpy would also form a 

 rope, and fall uniformly according to the same law, a' though there is 

 here no cohesion in force ; it is true from the mere effects of gravita- 

 tion ; but in order to keep up a continuous stream, if I may use the 

 expression, the successive shillings must be dropped from a height 

 such as to impart to them the velocity wddch the column has already 

 acquired. In this case there is no cohesion, which causes ' one shil- 

 ling to draw the one immediately above it,' and, therefore, why ad- 

 duce this property of water to explain a (act when it can be ac- 

 counted for by gravitation alone which acts on all bodies ? 



" I have already occupied so much of your valuable space, th.it I 

 must defer bringing forward many illustrations of the above truths 

 for some future period, hoping that you in the mean time will re- 

 member vour promise, and enlighten us on a subject at present in- 

 volved in so much obscurity. Since writing the above, I have met an 

 authority of such eminence and celebrity on subjects of this nature, 

 to support my views, that I cannot forbear quoting him at full length. 

 The authority I allude to is Belidor, in the edition of his 'Architec- 

 ture Hydraulique,' published at Paris in the year 1737, at page 170, 

 paragraph 429 and 430, it is stated, 'that when a vertical pipe, of 

 which the opening is equal to the base, is allowed to empty itself, the 

 surface of the water acquires in falling a velocity which increases as 

 that of bodies subject to gravity, which fall freely.' This is as you 

 stated. In the next paragraph, 430, he states, ' As it is always pos- 

 sible to render uniform a retarded or accelerated velocity, in taking 

 half of the greatest velocity, this must be done when we wish to 

 compare the discharge (la di-pense) of a pipe, such as the preceding, 

 with one always maintained full ;' and in paragraph 431, always al- 

 luding to pipes, ' the velocities of water are in the ratio of the square 

 roots of the heights ;' and further on he states that the velocity of 

 the issuing water in a pipe constantly full, is that due to the whole 

 height and not half; and also 'we can then, when it is convenient, 

 substitute for the velocity of the water of the column, the square 

 root of the height of the pipe.' I must conclude by requesting any 

 of your readers who are not satisfied with my arguments, to peruse 

 that part of Belidor from which I have quoted the above passages. 

 " I have the honour to be, Sir, 



" Your obedient servant, 



"T. F N." 



P.S. " In addition to the extracts which I have already given you from 

 Belidor, I beg to add the following, in order to snow more clearly that 

 I have his support to my statement that the velocity is that due to the 

 whole height. Chap. 3, paragraph 438, ' We can then say that the 

 discharge (la dfpense) of a pipe or reservoir during the time neces- 

 sary for a body to fall freely from the height of the surface of the 

 water above the bottom, is equal to a column of water which has for 

 its base the orifice, and for height a line equal to the space which a 

 body can move over with an uniform velocity during the time of the 

 fall "with the acquired velocity.' Apply this to a pipe of 16 feet ; the 

 time is a second, the acquired velocity 32 feet per second, and not 16 

 feet, as stated in your remarks. l Nothing can be more to the point 

 than the above extract. It is true that Belidor has not remarked that 

 the velocity of the whole column, if maintained full, increases every 

 second, although it would appear that he was aware of the fact, from 

 his statement in paragraph 573, when in treating of the momentum of 

 water issuing from an orifice, be states that a pipe can never be main- 

 tained full by a reservoir unless the pipe be of such small diameter, 

 that the friction retards the water more than gravity accelerates it. 



Placing out of consideration for the present, the minor point re- 

 specting the cohesion of fluid particles, we shall confine .ourselves 



« It was the mean velocity we stated to be 16 feet per second, and uot the 

 final velocity. 



