1843.] 



THE CIVIL ENGINEER AND ARCHITECT'S JOURNAL. 



141) 



water at the end of the 1st ft. is 7-3103 ft. per second, (and not 4 ft. 



as stated by your correspondent T. F n) ; at the end of the 4th ft. 



it is 15-6562 : and at the 16th ft. 31-3176 ft. per second : or, in round 

 numbers, 8, 16, and 32 ft. per second. If then the motion of the 

 whole of this column is uniform, whence has the power been obtained 

 to increase the velocity of water at the end of the first foot, from 8 

 feet per second, to, say, 16 or 32 feet per second ? Doubtless from 

 the superior velocity due to the lower parts of the water in the pipe. 

 If this is the case, then must the natural velocity due to the lower 

 portions of the water be diminished ; for nobody when in motion can 

 impart any motion to another body at rest, or increase the velocity of 

 that body if in motion, without a diminution of its own velocity, in 

 proportion to the respective momenta of the two bodies. — Therefore, 

 it is not possible that the water at the end of the 16 feet pipe, whose 

 natural velocity is 32 feet per second, can, after it has imparted a 

 quicker motion to the higher portions of water, still possess the 

 same velocity that it would have had, if it had not parted with a 

 portion of that velocity, in increasing the velocity of the upper parts 

 of this column. Consequently, it follows, that the velocity of the 

 water issuing from the pipe must, (if the stream fills the pipe and 

 flows uniformly through it), be diminished, and therefore be less than 

 32 feet per second. 



Next, as to the cause of the solidity of the issuing stream. — Let us 

 first consider the phenomena of a stream of any liquids falling from a 

 height, but not down the interior of a pipe ; or even a continuous 

 stream of leaden bullets. For a short distance we perceive the stream 

 solid, till the increased velocity of the lower parts as they descend 

 causes them to leave those immediately above them, air filling up the 

 intervening space, and thus the solidity of the stream is broken. Now 

 let us turn to the pipe, and we see, by the above table, that the natural 

 velocity of a layer of water which has fallen one foot, is 7-3103 feet 

 per second, and of another which has fallen two feet, 11-1)637 feet per 

 second; why then does not this lower portion fall faster than the 

 higher one, and separate from it, and thus break the solidity of the 

 stream ? Simply because if it did so, there would be a vacuum 

 between these two portions of water, bacause the sides of the pipe 

 are impervious to the air ; and as the atmosphere is pressing on the 

 water at the top and bottom of the pipe with a pressure, say, of 14 

 pounds per square inch, these portions are kept together by this 

 pressure ; for no sooner does an under stratum of water try, as it were, 

 to leave the stratum above it, and form a vacuum, than the presure 

 of the atmosphere at the top of the pipe is brought into action, and 

 the velocity of the upper stratum increased ; while at the same time 

 a portion of the atmospheric pressure, being by the same tendency 

 removed from the upper surface of the lower stratum, the. full pressure 

 of the atmosphere is exerted upwards at the base, and the velocity of 

 this lower stratum is consequently diminished. This certainly appears 

 to account more satisfactorily for the solidity of the stream, than the 

 force of cohesion; for this force must have the same influence over 

 the particles of water whether they are in the inside or on the outside 

 of the pipe. — As a proof, that when water is descending in the 

 interior of a pipe, which is continually covered with water on the top, 

 the atmosphere has a tendency to rush in through the sides. I need 

 only mention the wellknown fact, that if in a pipe under these circum- 

 stances there happen to be a crack, or a hole bored through the sides, 

 the air immediately rushes in, and the solidity of the stream is 

 destroyed. 



Lastly, to determine the actual velocity of this continuous column 

 of water. — Let us suppose the water in the pipe composed of an 

 immense number of laminae, eacli of the thickness of a particle or atom 

 of water, and consequently of the same thickness and weight; and let 

 «s take under our consideration the case of any two of these contigu- 

 ous laminae, and suppose that the velocity due to the higher one is, 

 say, one inch per second ; and the velocity due to the lower one is, 

 say, two inches per second. Then if these two lamina;, when moving 

 with these velocities, be at the same instant connected by the 

 pressure of the atmosphere, or otherwise, so that the one cannot move 

 on without the others moving along with it, it is evident, since the 



quantity of matter in each is the same, that the resultant velocity of 

 the two will be one half of the sum of their original velocities, or one 

 and a half inches per second. If then we could obtain the natural 

 velocities due to each respective lamina, the average would be the 

 result and velocity for the whole column. On taking the average of 

 initial and final velocities due to the water at the end of each succes- 

 sive two feet, from the foregoing table, and then dividing their sum 

 by 8, we have 20-6077 feet as the. average velocity of the whole 

 column. And if by approximating rather more closely to the system 

 of atomic laminae, we take the average of the initial and final veloci- 

 ties due to each_/W of water, and divide the sum by Hi, the resultant 

 is 20-7265 feet per second. 



Again, since " The velocity acquired by a body falling from rest in 

 free space is as the square-root of the space fallen through," the 

 space varies as the square of the velocity. — If then, we take the 

 spaces fallen through at the termination of each foot in the 16 feet, 

 (as in the first column of the above table) as abscissa, and the respec- 

 tive velocities (as in the second column) as ordinates, the resultant 

 curve will be a parabola similar to the figure below. — 



Let us suppose the line AB, represented in this figure 16 feet, or 

 the space fallen through, to be divided into an almost infinite number 

 of parts equal to the number of atomic laminae in the column of water; 

 then if we could find the length of the several ordinates at those 

 points, and divide the length of the whole by their sum, the 

 result would be the average length of all the ordinates, and con- 

 sequently the average velocity. This can be done by finding a 

 rectangle whose area is equal to the area of the parabola, and one of 

 whose sides is equal to AB. — Now, i x abscissa AB x ordinate BC 

 = area of the parabola = | X 16 X 31-3176 = 334-0544 ft. There- 



334-0544 8x16x31-3176 „__ Q . ., „ 



fore — = rj; = 20-8784 ; consequently the 



two sides of this rectangle are 16 feet and 20-8784 feet, of which 1(3 

 feet represents the space fallen through and 20-8784 feet the average 

 of all the ordinates, and consequently the average velocity. 20-S784 

 feet therefore is the average of the natural velocities due to the 

 several atomic laminae composing this column of water. Again, it is 

 evident, that since the abscissa AB (or 16 feet), enters both into the 

 numerator and denominator of the fraction representing the average 

 velocity, it may be eliminated altogether; and the expression then 

 becomes § X 31-3176; or, "The velocity of water descending the 

 interior of a pipe 16 feet long, is equal to § of the natural velocity 

 due to a body, after it has fallen through a distance, in free space, 

 equal to the length of the pipe." 



I. T N. 



Shropshire Lunatic Asyi.sw.— We understand Messrs. Cooper, builders, 

 of this town, have entered into contract with the magisirates of the county 

 of Salop, for the erection of a Lunatic Asylum for that county. The build- 

 ing is from a design by Messrs. Scott and Moffatt, of London, in the 

 Elizabethan style. The extent of the front 280 feet, and arching 170 feet, 

 The plan is in the shape of the letter 11. The first part of the building will 

 cost £11,000.— Derby Repi 



20* 



