ON THE FLOW OF WATEH THROUGH ORIFICES. 



263 



of the crest there will flow a quantity of water exactly proportional to the 

 length of E F if the width of the notch be varied while the depth h of the 

 water remains unchanged. Let the length E F bo denoted by I ; then 



Z = L — mh. 



Now, out of the entire flow, conceive the middle portion which flows over 

 E F, and may be regarded as bounded laterally by two veilical planes per- 

 pendicular to the plane of the orifice, one passing through E R and the other 

 through F S, to be taken away ; and suppose the two remaining parts which 

 flow over AE and BF, with the necessary lateral parts of the notch-plate, 

 to be brought together as shown in fig. 13, so as to form one notch having 



Fig. 12. 



mh for its width and h for the height from crest to still-water level, and in 

 which, therefore, the width of the notch shall bear a constant ratio to the 

 height of the water, when the height varies, the width being always m times 

 the height. 



Then, by exactly the same mode of procedure as that already 'used for 

 finding a formula to show how the quantity of water flowing in a V- notch 

 varies with the depth of the vertex or with any other linear dimension of the 

 flowing stream, we can readily see that if we put Q' to denote the volume of 

 water flowing per unit of time in the case represented in fig. 13, we shall 

 have 



Q' = a/tVA, (10) 



where a is a constant coefiicient. 



Next to find an expression for the quantity (voluminaUy reckoned) flowing 

 over the middle part E F of the crest, we may consider, first, of that middle 

 part, a portion G K taken always of a length bearing a constant ratio to h ; 

 and for simplicity we may take it of length equal to h*. Now in this stream, 



* Or, to meet the case in wliich tliere might not be, between E and F, a length so great 

 88 h, we might as well consider, in another notch having great width and having a height 

 of flow equal to h, a portion of the flow not near either lateral extremity of the notch, and 

 occupying a length oi the crest equal to h. 



