the ratio 32: 1. The author is given to understand that this ratio has been adopted as a matter of experience in the practical construction of machinery, independently of Wöhler. Launhardt quotes an older method of American engineers. According to this method, if tension alone alternating from 0 to 700 kilogrammes per square centimeter were admissibie, for alternations of tension and compression the following formula was used: . could be taken into account in the Italian edition of the work already frequently quoted. The following short demonstration may serve as a supplement to the English edition. Of course it is not a question of improving the very defective theory of buckling of Euler and Navier, still less of reconstructing the empirical formula of Rankine and Gordon. Nevertheless, the author is so much convinced of the inaccuracy of the latter makeshifts, that it appears incredible to him how some (26) English engineers, who remain contented with them, could raise objections on the score of inaccuracy to the formulæ given above. According to the preceding (X), if (27) Fa denote the requisite sectional area of a strut without liability to buckling, the stresses on which alternate between the maximum compression Ba and the minimum compression Bɛ, then For p 0,1, 1, 2, 1, according (27)b=700, 560, 477, 400, 350 to formula (23)b=700, 612, 525, 437, 350 Wöhler concluded from his own experiments that the following values were admissible for iron structures of unlimited duration: For alternations between tension or compression only, and an unloaded condition, b=1,100 (above, 700), for alternations between equal tensile and com pressive stresses, b=580 (above, 350). Previous to his becoming acquainted with the Weyrauch formula, Launhardt for Ba <Bt,Fa= Bi recommended of (26). a provisional adoption XI. ON THE LIABILITY TO BUCKLING. In the preceding demonstration, the buckling strength has not been considered. = Bd (1-n -N Bt Ba (29) Bi (30) b All stresses are to be substituted here as numerical values without signs. XII. CONSIDERATION OF THE LIABILITY TO BUCKLING. Other methods proposed in Germany and Austria have also left this question If for any strut or pillar, be the out of consideration,* as it was formerly length, F the sectional area, I the least almost universally the practice in those moment of inertia of the section for countries to calculate even the struts of axes through its center of gravity and lattice bridges for simple compression lying in its plane, E the modulus of elasonly. It was sought to avoid the risk of buckling by large moments of inertia of sections, and when necessary by fixing intermediate points. Since, how ever, the neglect of the liability to buckling had been noticed by American reviewers, the author first showed how it *Shäffer in his modification of Gerber's method has subsequently taken it into account. Deutsche Bau- and there results, according to the zeitung, 1877. theory, as mean stress per unit of sec- Navier's theory of buckling are only to tion at the moment of buckling : tk= be used for u>1, as for μ< 1 the calcuδΕΙ t lation for uniform compressive stress (32) only by (28)-(30) leads to greater secFI μ tions. Hitherto in determining dimenFor the transmission of a pressure Basions many engineers have been in the there is required with a static load, habit of combining the formulæ for uniwithout liability to buckling, a section form compressive stress and buckling, with liability to buckling, a section so that instead of writing in the one case without liability to buckling Ft= Ba and in the other with liability to buckling Ft BR==μBa, they introduced, generally, according to the practice of Rankine and Gordon, Bd Bd tk In a sectional area F, the greatest to buckling Let be the hypothetical stress per unit of area obtained on the supposition that the greatest stresses implied in (28) and (32) have the same ratio as the required sectional areas. Then Ba Bd Ba t tk Substituting from (32) there follows: Bd k= F t tk F This stress would be that due to a force equally distributed over the whole sectional area, (33) If B be substituted throughout in place of Ba, the same formulæ apply as without liability to buckling. The compress In an exactly corresponding manner the method of determining dimensions here under discussion may be proceeded with. Substitute, instead of in the one case Ba and in the other Ba, generally, (1+) Ba; whence instead of the formulæ (28)-30) applicable only for μ<1 and of those (34)-(36) admissible only for u>1, the following formulæ, always applicable to bars under compression, with or without liability to buckling, appear: for bars under compressive load only: ive force Bd, with liability to buckling, if (1+μ)Ba>Bt, F= The necessary sectional area on the assumption of a liability to buckling follows from (28)-(30): (40) These equations resolve themselves for u 0 into those applicable for simple compression only (28)-(30), and for very large as compared with 1 into the |formulæ (34)-(36) derived from theory of buckling. They give larger sections. than each of the corresponding others v (1 - "Ba) (36) respectively. nuBa Bt XIII. ANOTHER METHOD OF CONSIDERING THE XIV. ORDINARY PRACTICE WITH LIABILTY TO The method pursued in determining dimensions with reference to buckling The formula (34)-(36) derived from will, in most cases, be the following: In the first place Fa is determined for simple compression only by (28)-(30), the section is provisionally arranged accordingly, u is found by (31), and the formulæ (34)-(36) or (38)-(40) can then be applied. The struts of bridges have usually a spread-out (gespreizten) section, and in most cases it will be found that u<1, so that for those who adopt Navier's buckling theory, and therefore the formulæ (34)-(36), no liability to buckling exists, and the values of Fa remain as the final sectional areas. Those, however, who have hitherto used the empirical formulæ (37) in determining dimensions, may, with equal propriety, apply the new method by means of the equations (38)– (40). The increase of section from Fɑ to F, rendered necessary by the liability to buckling, may in struts of bridges be usually attained by a slight strengthen b=700 (1-2x40,000) 1 grammes. This coefficient would be calculated for the various methods of securing the ends of struts with reference to the Example 8.-What limiting stress per values of corresponding to the theory square centimeter should be chosen for of buckling from (31). At the same the booms of a lattice girder when the time the theory does not agree particu- ratio of the dead weight to the greatest larly well with experience; and, in practice, ends absolutely rigidly fixed, or total load amounts to: perfectly free to turn round the centers of their joints, by reason of frictionless By (25) articulation, do not actually occur. For b=700(1+ iron bars with so-called fixed ends, the author is in the habit of making 6= 24,000; for, iron bars with both ends 745. * secured by pins (American bridges) = 18,000 has been given.† =700 (1 = ? 1 2x3.5 800 kilogrammes. With 640 instead of 700 it would be Example 4.-The load on a hollow cylindrical column varies between the comFor the other constants occurring in pression Bd= 50,000 and the compresthe preceding formulæ v = 700, m=nsion Bs 20,000 kilogrammes. The = = *For this values from 10,000 to 36,000 are in use. With E-2,000,000, and t=3,300 kils. per square centimeter, the theory gives, according to (31), 7=23,900. Later American experiments (Civilingeineur, 1878, pp. 17-28) lead with the assumption of formula (37) to a mean value 22,700. + Prof. Cain in Van Nostrand's Eclectic Eng. Mag., 1877, p. 458. sectional area for the length = 400 cen *For further explanation for special structures, see Weyrauch, "Strength and Determination, &c.;" on the calculation of rivet joints, the same work; on the consideration of the liability to buckling Weyrauch, "Stabilita dele construzioni in ferro ed in acciaio;" on a special method of taking into account the shocks produced by the moving load, Zeitschr. d. Ostreich Ing. u. Arch. Vereins, 1879. 103.0 centi timeters is to be calculated with refer- To make up this section the thickness of ence to buckling. the angle irons may be assumed as 1.2 From (28) or (23) or Example 1, where centimeter, and then F = there is no liability to buckling, Fa = meters. 59-52 square centimeters. Correspond- If the calculation were repeated with ing to this, the section is temporarily the more exact values of F and I, there assumed as 1028, μ. a ring of 10 centimeters would result consecutively I = outer, and 9 centimeters inner radius, whence follows F = 59.7, I = 2,700 centimeter-units. Equation (31) gives = 0.17, F = 101.82. Consequently a value of F has been determined to small by 0. 68 square centimeter = per cent. an error that may be neglected. Example 6.-To calculate the strut given in the preceding example for the length = 300 centimeters. In this case the provisional values of F and I are exactly as above; on the other hand μ = 0.40, and since (1+μ) Ba= 42,000 > Bt, by (39) According to the theory there is there- fixed. XVI. CONCLUSION, centim. Square centimeters. This could be attained by keeping to the outer radius of 10 centimeters, by reducing the inner It is possible, as already indicated, to radius to 8.84, giving F = 68.6. find many deficiencies in the method Example 5.-The stress on a strut in demonstrated. But, there are here only the web of a bridge girder alternates be two questions for consideration; (1) Does tween the tension Bt = 40,000 and the the method lead to the expectation of compression Ba = 30,000 kilogrammes. greater security than that hitherto used The sectional area for the length = 200 in determining dimensions? (2) Is not centimeters, with reference to buckling, In order to facilitate in the latter paranother newer method to be preferred? is to be determined, the ends being ticular the judgment and independent From (30) or (23), and as already shown choice of his English professional felin Example 2, without liability to buck- low-workers the author will be ready to ling, Fa= 91-43 square centimeters. give in a second Paper, a short explanation and comparison of all the methods hitherto proposed. His object is mainly to supersede the present method of deEven if it is termining dimensions. thought that other formulæ for a and b should be preferred, the general principles here demonstrated may yet have contributed to the elucidation of the question. The strut is constructed in the manner of a plate girder, of four angle-irons, 92 × 2002 852×24,000 =0.18, so that here, too, by the theory there is no liability to buckling. If, however, the empirical formulæ are to be used, then from (40), since (1+ μ) Ba= 35400 < Bt. The Since the strength of materials has recently acquired a fresh interest, it may well be that at no distant time satisfactory relations that will admit of general application may be discovered. author would welcome them with pleasure. It is sufficient for him to have contributed towards the supercession of the 182.5 sq. centim older method of determining dimensions so as to make way for a more rational system. THE AMOUNT OF RAINFALL IN ITS RELATION TO THE WATER SUPPLY OF A CITY. By JULIUS W. ADAMS. Written for VAN NOSTRAND'S ENGINEERING MAGAZINE. AN article under the above heading the water shed to determine the amount appeared in the last number of the Maga- available." Subsequently, making an zine, upon which I would beg leave in allowance of 60 per cent. for the annual the interests of the younger members of evaporation from reservoir surfaces, and the profession, who consult your pages obtaining a monthly mean and ratio for with profit, to make a few brief com- this, finally embodies the whole in a table for use, "in the calculations upon ments. The true theory of any subject, must, water-works." of course, embrace all the facts known If the writer had confined himself to of it. Thus far we have too few facts the phenomena attending the local rain upon which to build a comprehensive fall, the deductions drawn would have system—not to say-science of meteoro- been interesting to the extent of showing logy, and some time will elapse before what had taken place, and what might we can pretend to reduce to rule the be anticipated in the locality chosenmany anomolous phenomena attending Troy for instance; but as he does not the rain fall, and the resulting drainage state explicitly, that his deductions are from the water shed. to be taken with great caution when used The writer of the article in question, in estimating a city water supply, but after a review of the rain-producing if I understand the scope of the article, processes over varied surfaces of the it is, that it offers a method by earth, and enumerating certain general which, at least in the Atlantic States, principles which are presumed to affect and to the extent of 100 square the precipitation of moisture, takes the miles of shed, the amount of water average annual rain fall in sections of derivable from an estimate of the mean the United States, and dividing these annual rains, may be used safely in essections into nine groups, obtains what timating the water supply of cities. It is characterized as a type curve for each is this which I take exception to-the group, representing the fluctuations of application of any general formula for the monthly rain fall; and derives from such purpose. However closely the rethe average rain fall of a series of years sults of using this method may approxthe factor 0.8, as representing the ratio imate the truth in particular cases, as of the rain fall for the dry years. Tak- shown by existing works already estabing a twelfth of the annual rain fall for lished, and that it may do so is unthe monthly mean, he determines from doubted, yet it cannot be assumed, a the record of averages of the Atlan- priori, as applicable in the future for tic coast stream flows, the ratio of the other localities. The whole principle of monthly flow of the streams to this mean estimating from yearly averages of rain as unity, and upon the supposition that fall has been proven by experience to 50 per cent. of the annual rain fall is col- be misleading, and, for the purposes of lected in the streams, and 0.8 represents a permanent city supply, no longer rethe ratio of a series of dry years to the commended. mean annual rain, obtains a mean monthly rain fall, which combined with the monthly ratio of the flow of the streams above noted, gives the available flow in inches monthly, and adds "which quantities have only to be multiplied by the number of acres or square miles in Much misapprehension exists as to the amount of water which may be calculated upon with certainty on a given water shed; and disappointment in many notable cases, which might be cited, has arisen from taking as a standard for computation the average rainfall of a series |