26,'J REPORT 187(). 



coutraction directly proportioual to the height of the stiil-wuter surface-Ie\cl 

 above the crest in order to lind what may be treated iu the formula as the 

 effective length. 



The formula in its general form here last noted expressed only in symbols, 

 as also in its subsequently developed form here previously stated with 

 numerical coefficients aiTived at by tentative application of numerous exjje- 

 rimenta, is thus to be regarded as an ingeniously arranged and valuable 

 empirical foimnla, but not as one founded on any trustworthy hydrokinetic 

 theory. It is founded partly on the old ordinary false " theoretical" views, 

 and partly on good conjectural assumptions, and is adjusted and approxi- 

 mately verified by elaborate experiments conducted on a scale unusually 

 large, and with unusually good means for attainment of exact results. 

 Mr. Francis, it is to be noticed, explains that, in the formula as finally 

 brought out, the index for the power of the height of the water is taken as 

 an exact fraction, %, in preference to some unascertained fractional expression, 

 different in no great degree from 4, merely for the attainuient of faeilitj" in 

 calculations in the practical applications of the formula, and riot for any 

 theoretic reason. Also it is to be noticed, in respect to the value -^ which 

 he assigned for the symbol h, that the symbol itself was tirst assumed as a 

 constind rather than some unknown variable dependent on /;, and was after- 

 wards fixed at the particular value J^f for the sake, in both cases, of attaining 

 a convenient degree of sim])licity which by trials was found to be attainable, 

 consistently with good accordance between the representations afi'orded hy 

 the formula and tiie results shown by experiments. He supposed, however, 

 that " many other values of a and b (probably an unlimited number) might 

 *' be found that would accord somewhat nearer with the experiments"*. 



Many years ago, after my having become acquainted with the empirical 

 formula thus made out by Mr. Boyden and Mr. Francis, it occurred to me as 

 desirable to attempt to investigate by hydrokinetic principles, without special 

 experiments, a true formula for the flow of water in rectangular notches in 

 vertical thin plates, or vertical plane faces, on the hypothesis of the water 

 being a perfect or frictionless lluid, and by using iu the formula symbols 

 for constant coefficients, which, after the finding of the formula, might be 

 determined by a small number of accurate experiments, and might further 

 be tested as to their trustworthiness, or might be amended so as to become 

 more exact, by a large number of varied experiments. It will be interesting 

 to notice that the formula which had previously been arrived at in America 

 by Mr. Boyden and Mr. Francis in the way already described is in perfect 

 agreement with the formula which, by ray own investigation, is brought out 

 by strict scientific principles as a highly exact formula for water considered 

 as a perfect fluid, and as being a very satisfactory representation of the truth 

 for real water. 



It is to he noticed at the outset that obviously a notch may be made so 

 long relatively to the depth of its crest from the still-water surface-level, 

 that, for any additional length, the increase of the flow will be proportional 

 to the additional length. Let niJi, in which m is a constant multiplier, be 

 such a length as that, for additional length, the additional flow ^ndll be pro- 

 portional to the addition made to the length. In fig. 12 let A B be the crest 

 of the notch, and let C D be the level of the still-water surface of the pent-up 

 water. Let AE and BF be each equal to |w7i, so that, over the part EF 



* Lowell Hjclvaulic Experiment?. § 1.56. p. 118; ^ 1 '->o, p 1 Ifi ; and tlie passigp quoted 

 above from f 123, p. 74. 



