March 6, 1890] 



NATURE 



411 



French popularizers of science, which has made them 

 masters of their art. 



The above-named volumes are three of a number of 

 similar treatises which have lately appeared. The ap- 

 preciation of the beautiful and generally interesting in 

 Nature must always precede the study of the more useful 

 and special, and it is the highest function of works like 

 the present to awaken this preparatory appreciation. Of 

 such works those are the most valuable whose authors 

 can claim a sound elementary knowledge of the facts with 

 which they deal, and a familiarity with current research. 

 Only on these terms can a popular natural history rise 

 above the level of the too well-known type, in which the 

 scissors supply the knowledge and the paste usurps the 

 place of the co-ordinating intellect. G. B. H. 



A GENERAL FORMULA FOR THE FLOW 

 OF WATER. 



A General Formula for the Uniform Flow of Water 

 in Rivers and other Channels. By E. Ganguillet 

 and W. R. Kutter. Translated from the German by 

 Rudolph Hering and John C, Trautwine, Jun. (London : 

 Macmillan and Co., 1889.) 



THE general formula devised by Messrs. Ganguillet 

 and Kutter for caculating the flow of water in both 

 large and small channels, under varied conditions, was 

 brought under the notice of English-speaking engineers 

 by the publication, in 1876, of a translation by Mr. 

 Jackson of some articles on the subject written by Mr. 

 Kutter, which appeared in the foiirnal der Cultur- 

 Ingenieur in 1870. This translation, however, was not 

 authorized by Mr. Kutter, and contained some incomplete 

 tables inserted by Mr. Kutter in his articles at the request 

 of a friend. The present volume is a translation of the 

 second edition of the treatise on the formula, written by 

 Messrs. Ganguillet and Kutter, engineers in Berne, who 

 have added a preface to the translation. Mr. Kutter died 

 whilst this translation was in progress ; and a short 

 memoir of him, with a list of his works, is appended to 

 the translators' preface. 



The book commences with an historical sketch of the 

 attempts to arrive at a formula for the flow of water in 

 open channels; and the insufficiency of the earlier formulae 

 is pointed out. The investigations of Messrs. Darcy 

 and Bazin, and the gaugings of the Mississippi by Messrs. 

 Humphreys and Abbot, are then concisely described, and 

 the formulas which they deduced from the results of their 

 experiments are given, the history of the subject, in a 

 brief form, being thus brought down to the period at which 

 Messrs. Ganguillet and Kutter commenced their investiga- 

 tions. This forms a sort of introduction to the account 

 of the conception and development of the general formula, 

 of which the various steps are described in detail. The 

 modifications for various amounts of roughness are classi- 

 fied ; and, finally, the formula is tested by the comparison 

 of its results with a number of gaugings under very differ- 

 ent conditions ; and these results indicate, in considerably 

 the greater number of cases, a closer approximation to 

 the actual measurements than those obtained with the 

 formula of either Humphreys and Abbot, or Bazin. A 

 supplement gives a more direct derivation of the formula 



for mathematical readers ; and the appendices contain 

 numerous tables giving the flow of water in pipes under 

 pressure, as well as in open channels, for practical use in 

 English measures, derived from the formula, and also a 

 diagram for the graphical determination of the values of 

 the factors in the formula, adapted to English measures 

 by the translators. 



Most of the hydraulicians who had investigated the 

 question before Darcy and Bazin, such as De Prony, 

 Dubnat, Eytelwein, D'Aubuisson, Downing, and others, 

 agreed in adopting a formula of the form V = t'v/RS, 

 of which Brahms and Chezy are said to have been the 

 authors in the latter half of the last century, in which V 

 is the velocity, R the hydraulic radius, and S the slope. 

 Different values were assigned to the factor c by the 

 various investigators ; but it was always regarded as a 

 constant, applicable to any sized stream in most cases, 

 to any slope, and to any state of the bed. Mr. Darcy 

 was the first who directed attention to the influence the 

 condition of the sides of channels and pipes exercised on 

 the discharge ; and he instituted a series of experiments, 

 carried out after his death by Mr. Bazin, by which the 

 flow of water in regular uniform channels, under different 

 conditions of slope, form, and roughness of bed, was 

 measured by careful gaugings and gauge-tubes. A few 

 years previously, Messrs. Humphreys and Abbot had 

 carried out their well-known gaugings of the flow of the 

 Mississippi by means of double floats, and deduced a 

 formula for the results obtained. Messrs. Ganguillet and 

 Kutter found that the formula derived from the Missis- 

 sippi experiments, relating to a large river with a very 

 slight slope, was not applicable to the small streams with 

 steep slopes of which they measured the flow in Switzer- 

 land, and also that Mr. Bazin's formula was not suitable, 

 in its original form, for large rivers with irregular beds. 

 This led Messrs. Ganguillet and Kutter to search for a 

 formula applicable to very different slopes and sizes of 

 channel, and adaptable to various conditions of bed. 

 They took as the basis of their formula the various ex- 

 perimental results obtained in France and America, 

 together with their own independent observations on 

 channels with steep slopes, so as to include the extreme 

 varieties of flow within the range of a single formula. 



Starting from Mr. Bazin's formula, V = / - 



+ 



/3' 



where c 



v„;i' 



r 



they eventually found it expedient 



to express the value of c in the form 



JL 



I + 



in which. 



VR 



though they at first assumed j and x to be constant for 

 any given state of bed, they finally modified them to 

 expressions varying with the slope. The alterations in 

 the formula were effected by aid of graphical representa- 

 tions of the various sets of gaugings. It was found, in in- 

 vestigating the various experimental results,that the factor t' 

 varied generally with the slope ; but a somewhat anomalous 

 result was also noted — namely, that whereas in the Missis- 

 sippi observations c increased with a decrease in the 

 slope, it on the contrary decreased with a decrease of 

 slope in the gaugings of small channels, unless the wetted 



