ON THIS FLOW OF WATKH THROUGH OKIFICES. 



265 



Fig. 14. 



Fig. 15. 



Also from (11), by changing, as done before, the letter /3 into the English 

 letter a, we see that for a notch with no end contraction (contractions being 

 prevented at both ends by vertical guiding side faces ijerpendicular to the 

 plane of the notch) we would have 



Q,=aUi' (14) 



Now the thi'ee formulas (12), (13), and (14) may be combined so as to be 

 expressed together, thus : — 



Q,=a{lj-nhh)h'-, (15) 



where n is the number of end contractions, and must be either 2, 1, or 0. 



To determine the constants a and 6, all that would be necessary would be 

 to have two very accm-ate experiments on the flow of water in one notch at 

 different depths, or in two notches of the same kind with the ratio of the 

 width to the depth not the same in both. Then, putting into the formula 

 the measured values of L, h, and Q for the one experiment, and then again 

 those for the other, we would have two equations with two unknown sym- 

 bols, and so we could And the numerical values of those symbols. It would, 

 of course, be desirable, for experimental verification of the theory on which 

 the formula is foimded, as also for mutual verification or testing of the expe- 

 rimental results themselves, to have numerous experiments on the flow for 

 various depths in various notches of difi'erent widths, so as to find whether 

 the formula would fit satisfactorily to them all, or to all of them that, after 

 comparison, would be found trustworthy — provided that the width of the 

 notch be not too small in proportion to the depth of the flow, or that in all 

 cases the width be sufficient to allow of there being at least some small part 

 in the middle where the rate of flow per unit of time would be proiiortional 

 to the length of the part of the crest to which that flow would belong. 



Mr. Francis's experiments and his reductions of the results carried out in 



