pipes varies, as the 2.5 power of the diameter (28), thus 42'5 = 32, and we shall therefore require 32 1-inch pipes to deliver with the same head and length the same quantity of water as a 4-inch pipe, and we may admit that a 4-inch main would supply 32 1-inch lead services, &c. Table 6 is calculated on these principles. TABLE 6.-SERVICE MAINS for WATER-SUPPLY in TOWNS. "General Laws for Pipes."—The following general statement of the laws governing pipe questions may be useful: some of these laws apply strictly only to long mains in which the head. due to velocity may be neglected. (27.) When d and L are constant, the discharge, or G, varies directly as the square root of the head, so that for heads in the ratio 1, 2, 3, the discharge would be in the ratio √1, √2, and √3, or 1, 1.414, and 1·732. Conversely, the head is directly as the square of the discharge, so that for discharges in the ratio 1, 2, 3, we require heads in the ratio 1o, 2o, 3o, or 1, 4, 9, &c. (28.) When H and L are constant, the discharge is directly as the 2.5 power of the diameter; thus with diameters in the ratio 1, 2, 3, the discharge will be in the ratio 125, 225, and 3o5, or 1, 5.6, and 15.6. Conversely, the diameter will vary directly as the 2.5 root of the discharge; thus for discharges in the ratio 1, 2, 3, the 2.5 diameter will vary in the ratio 2/1,2/2, and 2/3, or 1, 1·32, and 1.55, &c. (29.) When G and L are constant, the head will be inversely as the 5th power of the diameter; so that for diameters in the ratio 1, 2, 4, the heads will be in the ratio 45, 25, and 15, or 1024, 32, and 1. Conversely, the diameter will be inversely as the 5th root of the head; thus for heads in the ratio 1, 2, 4, the diameters would be in the ratio 4, 2, and √ī, or 1·32, 1∙15, and 1.0, &c. (30.) When H and d are constant, the discharge will be inversely as the square root of the length; thus for lengths in the ratio 1, 2, 4, the discharge would be in the ratio√4, No√2, and √1, or 2.0, 1.414, and 1.0, &c. Conversely, the length varies inversely as the square of the discharge; thus for discharges in the ratio 1, 2, 4, the lengths would be in the ratio 42, 22, and 12, or 16, 4, and 1, &c. (31.) When G and d are constant, the head is directly and simply as the length; thus for lengths in the ratio 1, 2, 3, the heads would also be in the ratio 1, 2, 3, &c. (32.) "Head for very Low Velocities."-Table 3 gives the greatest possible facility for the calculation of pipe questions, as may be seen by the examples we have given, and for all ordinary cases the results are correct; but for very small velocities with low heads, say under one foot, &c., experiment has shown that the discharges are less than that Table would give, and for such cases Prony's more difficult and laborious rule seems to give the most correct results. The following rule is based on that of Prony: Thus, say we required the discharge by a 12-inch pipe 3000 feet long with 36 inches head: then 16.353 x 36 × 12 427.4 gallons. +00665 = We may compare this result with that by Table 3, or rather by the rule ((3 d)° x H) + × L = G, given in (5), by which the discharge comes out 426 gallons, or practically the same as by Prony's rule. With a very small head, however, the two rules do not agree; thus, with only one inch head, this same pipe gives 54 87 gallons by Prony's rule, whereas the other rule gives 70.98 gallons, or 29 per cent. more. With a large head, on the contrary, Prony's rule gives a rather larger discharge than the other. The general comparison of the two rules may be shown by the case of a 10-inch pipe, 1000 yards long, the calculated discharge of which, with different heads, is given by the following Table : (33.) When the head is the unknown quantity, and the rest of the particulars are given, the rule becomes : Let us take an extreme case, in order to illustrate more fully the special adaptation of Prony's formula to very low velocities. Say we require the head for a 10-inch pipe 4000 feet long, discharging only 20 gallons per minute: then 3, the head comes out 00001646 × 1333 == 263 inch only; so that in this very extreme case Prony's rule gives in (5) or Table 3. • 626 = 2.38 times the head by the rule (34.) Table 29 has been calculated by the following modification of Prony's rule : In which d = diameter of pipe in inches. V = velocity of discharge in feet per second. H = head of water in inches. Table 29 has been calculated for small velocities only, because Table 3 gives results sufficiently correct for practical purposes, with higher velocities, and is more facile in application. We have added opposite each velocity in Table 29 the corresponding discharge of pipes, from 1 inch to 24 inches diameter, in order to abridge the labour as much as possible. For the use of this Table we have the following rules : (35.) 1st. To find the discharge, having H, L, and d given. Multiply the given head in inches by the diameter in inches, and divide by the length in inches, and find the nearest number thereto in Col. 1. Then opposite that number, and under the given diameter will be found the discharge in gallons per minute Say, we take the case in (32) to find the discharge of a 12-inch pipe 3000 feet or 36,000 inches long, with 36 inches head. Then H x d 36 × 12 = 012, the nearest number to which in D Col. 1 is 01192, opposite to which, and under 12 inches diameter, is 427 gallons, the discharge sought. 2nd. To find the head, having G, L, and d given. In Table 29, under the given diameter, find the nearest number of gallons, and take from Col. 1 the number opposite to it, which number, multiplied by the length in inches, and divided by the diameter in inches, will give the required head in inches. Thus, taking the extreme case in (33) to find the head for a 10-inch pipe 4000 feet long, with 20 gallons per minute:-The nearest discharge under 10 inches diameter is 20.45 gallons, opposite which in ·0001341 × 48000 Col. 1 is 0001341, and from this we obtain 10 = 643 inch head: the exact head for 20 gallons we calculated in (33) to be 626 inch. It should be observed that Prony's formula does not include the head due to velocity of entry (12), which for short pipes becomes important. It has been omitted in the preceding illustrations, because with such long pipes as were given in our cases it is too small to affect the result sensibly: for instance, in the last case, the head for velocity with 20 gallons per minute 20 2 and a 10-inch pipe by the rule in (3) is = ⚫000237 100 x 13 foot, or and of an inch only. (36.)" Square and Rectangular Pipes."-The case of square or rectangular pipes may be assimilated to that of round ones, and the head or discharge may then be calculated by the same rules and Tables that we have given for the latter. The velocity of discharge, whatever may be the form of the pipe or channel, is proportional to the hydraulic radius (57) or the sectional area, divided by the circumference or perimeter: in round pipes this is always equal to one-fourth of the diameter. Say we have a rectangular channel 3 ft. × 1.5 foot, Fig. 39; the area is 4.5 feet; the perimeter 9 feet, and the hydraulic 4.5 radius = 5 foot, which is the same as that of a round pipe 9 5 x 4 = 2 feet diameter. Then to find the head for friction |