VOL. L.] PHILOSOPHICAL TRANSACTIONS. iQS 



Let b = breadth of the channel in feet. 



c = breadth of the water-way between the obstacles. 



V = mean velocity of the water in feet per sec. 

 Now 25 ; 21 : : c : -f^ the water-way contracted ; byprincip. 3. 



And -^ c '. h : : V '.■—- V the veloc. per sec. in the water-way between the ob- 

 stacles ; by princip. 5. 



Also (2 ay '. vv : : a '. — the height fallen to acquire the vel. v; by 1 and 2. 



And (2 ay : (^ )* X vv : -. a -. (^)^X — the height fallen to acquire the vel. 



-— ?;; by 1 and 2. 



21c ' -^ 



Then (--)" X is the measure of the fall required; by 7. 



^21 c' 4rt 4a T ' y ' 



Or \_{^y — l] X — is a rule, by which the fall is readily computed. 



Here a = 16,0899 feet, and 4a = 64,3596. 



EXAMPLE I. For London Bridge. 



By the observations made by Mr. Labelye in 1746, 



The breadth of the Thames at London bridge is 926 feet. 



The sum of the water-ways at the time of the greatest fall is 236 feet. 



The mean velocity of the stream taken at its surface just above bridge is 3-f 

 feet per second. 



Under almost all the arches there are great numbers of drip-shot piles, or 

 piles driven into the bed of the water-way, to prevent it from being washed away 

 by the fall. These drip-shot piles considerably contract the water-ways, at least 

 •^ of their measured breadth, or about 3g^ feet in the whole. 



So that the water-way will be reduced to 1964^ feet. 



Now b = 926 ; c = 1961- ; v = 3-^ ; 4 a =: 64,3596. 



rj^, 25b 23150 ^ «^^„« 



Then - = ^^= 5,60532. 



And 5,60532=^ = 31,4ig6; and 31,4196—1 = 30,4196 = (1^)" — I. 



Also VV = (-^y = ->. ; and — = -7^ 'd—7rr;r^ = 0,15581. 



Then 30,4196 X 0,15581 = 4,739 feet, the fall sought after. 



By the most exact observations made about the year 1736, the measure of the 

 fall was 4 feet 9 inches. 



EXAMPLE II. For JVestminster Bridge. 



Though the breadth of the river at Westminster bridge is 1220 feet ; yet, at 

 the time of the greatest fall, there is water through only the 13 large arches, 

 which amount to 820 feet : to which adding the breadth of the 12 intermediate 

 piers, equal to 174 feet, gives 994 for the breadth of the river at that time: and 



c c 2 



