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MBER 28, 1899. CONCERNING LAKE LEVELS. _ ‘The correction of the natural outlets of the lakes and the construction of artificial ones has raised the question as to their effects upon lake levels, the depth of harbors, etc. The Great Lakes may be compared to four large water tanks at different elevations, sideways, above one another, each tank provided for carrying off the surplus water, with _ aconduit reaching into the next lower tank, the lowest of _ them discharging its surplus water into a ditch. Besides a continual supply from extraneous sources, the supply for each tank is increased by the surplus water from the next higher tank, keeping conduits filled up toa certain point, the water stages varying with the supply. If, in an arrangement as described, one of the tanks be- comes leaky, or any of its contents is diverted into a new _ channel not emptying into the old system, all points of the system below the point of leakage or tapping will suffer a diminution of the supply which, as a consequence, will _- lower the previous water stages. According to Henry Law, civil engineer, the velocity in channels of the same depth and!width throughout, equals 91.44 multiplied by the square root of the product of the sectional area and the tangent of inclination of the water _ surface, divided by the square root of the wetted perimeter see Mathematical Tables, by Henry Law and Professor oung, page 53). As the volume of water carried off equals sectional area multiplied by velocity, the volumes for differ- ent water stages are as the square roots of the third power the sectional areas, the small difference in the wetted perimeters and in the inclination being neglected. There- re, if the volume of water for a certain stage is known, it is easy to find it for another stage not widely differing from the former. According to the preceding formula the fol- lowing table shows, in figures, the relations between in- creased sectional areas and discharges : TABLE I. Sectional Area.} Discharge. A 1.00 1.000 ek ed : 1.05 1.076 I.10 I.151 1.15 1.233 1.20 1.315 1.25 1.398 This table shows, for instance, that, if the sectional area is increased 20%, the discharge increases. 31.5%, etc; and that the discharge increases faster than the area of cross section. The effect of dredging is hereby illustrated. For rivers of irregular shape and depth, the cross sections “of least area determine the outflow, d viding the river into sections of different lengths, with diffcrent velocities, but the discharges of the different sections may be assumed to be alike and to follow the same law as in channels of regular shape. The tapping of the Great Lakes by the Chicago drainage canal affects directly Lake Michigan, Lake Huron and Georgian Bay, all being of thesame level. According to -Thompson’s Coast Pilot those waters represent an area of 22,000, 21,000 and 5,000 square miles, respectively ; in total, 48,000 square miles. ' The cubic contents of a body of water of one square mile one inch deep equals 5280-5280 = 2,323,200 cubic feet, con- 12 - sequently 48,000 square miles one inch deep represent 2,323,200 X 48,000 == II1,513,600,000 cubic feet. _ The discharge of the drainage canal is said to be 10,000 cubic feet per second. A day of 24 hours equals 86,400 sec- onds, and a year has 365 times as many seconds. Hence, he discharge per year will be 86,400 365 X 10,000 = 315,- - 360,000,000 cubic feet. Therefore, the depth required to offset this yearly drain is 315,360,000,000 = 2.83 inches, III,513,600,000 in round figures, 3 inches. ‘But assuming the discharge to be 17,000 cubic feet per second, as some people have it, the depth required will be _17,000X 2.83 = 4.811 inches, or nearly 5 inches, by which 10,000 : _ the surface of Lake Michigan and Lake Huron will be per- anently lowered. To determine the effect which a lower level of Lake Huron will have upon Lake Erie and its connecting link, the formula mentioned above may be transformed by sub- stituting depths for areas, because the sectional areas are THE MARINE RECORD. nearly as the depths. Thus, we find, that the discharges are as the square roots of the third powers of the average depths. By average depth is understood the average depth of a cross section of least area. As the discharge of Detroit river is said to be 230,000 cubic feet per second, the decrease of the discharge on account of a lower level of Lake Huron is easily found. This decrease would diminish the supply of Lake Erie and affect its water level over its whole area of about 10,000 square miles toa certain depth, if not counter- acted by the increase of head, as illustrated by the following table : TABLE II. I 2 3 4 5 6 Par Average depth of | Discharge |Decrease of discharge ceag : cross sectionat present! of 1 of Detroit river. g E FA ee _ reduced 5 in. reduced to $8 98 Per cent. pee a $8 Ha A 10 oy Ae 0.9382 6.18 14,214 19.30 II Io 7 0.9437 5.63 12,949 17.58 12 117 0.9484 5.16 11,868 16.11 13 T2237 - 0.9523 4.77 10,971 14.89 14 13% 0.9557 | 4-43 10, 189 13.83 15 14 7 0.9586 | 4.14 9,522 12.93 16 15.7 0.9612 3.88 8,924 12.11 17 16 7 0.9635 3.65 8,395 11.40 18 ees 0.9655 3-45 75935 10.77 19 Popeyes 0.9673 3207 7,521 10.21 20 TOE ey 0,9689 | 3.11 7,153 9.71 Column 6 is obtained by multiplying the figures in column 5 by 0.0013574 = 86,400%365 Column 6 shows how 2, 323, 200 X 10,000 many inches the level of Lake Erie would be lower if the volume of its present outflow remained unaltered. But asa lower level diminishes the outflow and increases the head, and, consequently, the supply, the level would risé again long before the lowest stage were reached,and the defficiency partly be made up, as it were, by suction from Lake Huron. The figures in column 6, however, are of further interest as they show at what stage the discharge of the connecting link will equal the present or original discharge. The co- incidence of these figures with those further on in Table III, column 3, is surprising and proves that the area of Lake Erie (10,000 square miles) answers exactly the requirements of a fixed relationship between the two lakes. If, in the fundamental formula mentioned above, the ‘head’? divided by the length of the conduit is substituted for tangent of inclination, the discharges are as the square roots of the product of the third power of the average depth and the head, from which, by transformation, are found the following neat expressions : Let x and y equal the lowering of the surface of Lake Huron and Lake Erie respectively, e equal the average depth and h equal the original head. Q equal the original dis- charge, and R the altered discharge on account of a lower level, then is: (1) QR. (ght e) a ey Q = 2eh (2) O-R 3x when x=y Gi ae. (3) y=(3h+1) x when Q—R=O (4) ye Ghee RO e When x = 5 inches and e = 8 feet, the difference in level between Lake Huron and: Lake Hrie (formula 2) transforms into : (2a) Q—R_ 5 QO 8e formula (3) into: (3a) y=10+5, (3b) y= 120 + 5 Sees e And formula (4) into: (4a) y—x=10, I20 (44) y= x= e : e In (3b) and (4b) y and x are expressed in inches when e in feet. y — x is the difference in the ‘‘ head.’ \) TABLE III. I 2 3 4 e Whenx=—y | WhenQ—R=o]| WhenQ—R=o Feet. and Q=100. |Krie level lower.| head increases. ORs Inches. Inches, Io 6.25 17.00 12.00 II 5.68 15.91 10.91 12 5.21 15.00 10.00 13 4.81 14.23 9.23 14 4 46 13.57 8.57 15 4.17 13.00 8.00 Iu 3.91 12.50 7.50 17 3.68 12.06 7.06 18 3.47 11.67 6.67 19 3.29 11.32 6.32 20 313 II.00 6.00 A comparison of column 2 in the preceding table with — column 4 in Table II shows a pretty close agreement, as it should be, because the head in both cases is considered con- — stant. When Q —R=o0, R=Q, that is, Lake Erie level. has fallen and its head been increased to such a point where the supply from Lake Huron, respectively, Detroit river will have attained its original magnitude, all under the assumption of the discharge of Lake Erie not being affected and remaining constant. . As there is reciprocity between height of level and dis- charge there must be a point or level in a water basin where — supply and discharge compensate one another and have come to an equilibrium. As the discharges for a level sur- face are as the square roots of the third power of the aver-. age depth we obtain by transformation, the difference of discharge equal 3y, y representing the difference in level 2d : and d the average depth of the controlling cross section. Hence, to balance the supply and discharge, we have the following expression : 3y x (3h+e)—ye [See formula r]. 2d 2eh From which follows, that =d(zh+e)x e (3h + d) : Therefore, y is a constant quantity. x Assuming x = 5 inches; d= 18 feet,e = 12 feet and h = 8 feet. : y = 18 (24+ 12) 5 12 (24 + 18) = 18. 36. 5 12. 42. = 3. 3. 5” 7 = 45 =6.43 inches 7 In other words, Lake Erie level will be lowered 6.43 inches when Lake Huron level is lowered 5 inches, provided the controlling sections at present are on an average 18 and 12 feet deep, respectively. As regards the Chicago drainage canal, its great defect is, that no gates are provided for shutting off the lake water. This neglect threatens not only the country at the lower end of the canal with inundation, but also threatens Lake Michigan and its connecting waters to be drained several feet, ifa break in the canal should happen. Such a calamity would paralyze the whole lake traffic. In view of the enor- mous interests at stake, it is about time that the federal government be petitioned to secure such safeguards as will regulate the outflow of the canal, and prevent disaster to spread over the whole country. JOHN MAURICE, Civil Engineer and Nautical Expert. Chicago, Sept. 26, 1899. i oO oe MARINE INVENTIONS. Patents on marine inventions issued Sept. 26, 1899. Re- ported specially for the MARINE RECORD. Complete copies of patents furnished at the rate of ten cents each. : 633,705. Sail-rig for ships. A. V. Smith, San Francisco, California. ‘ 633,811. Bascule lift-bridge. J. P. Cowing, Cleveland, Ohio. tsa 633,873. Apparatus for cleaning hulls of vessels. Davi Mason, New York, N. Y. 633,903. Buoyant propeller. Mass. 633,904. Ballast device. J. P. Pool, New York, N. Y. 633,910. Machine for cleaning ships’ bottoms. John Schnepf, New York, N. Y., assignor of one-half to W. C. Doscher, same place, G. H. Pond, Ashburnham,