Here _a_ = 16,0899 feet and 4_a_ = 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 3⅙ 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 ⅙ of their measured breadth, or about 39⅓ feet in the whole. So that the water-way will be reduced to 196⅔ feet. Now _b_ = 926; _c_ = 196⅔; _v_ = 3⅙; 4_a_ = 64,3596. Then 25_b_ ⁄ 21_c_ = 23150 ⁄ 4130 = 5,60532. And 5,60532² = 31,4196; and 31,4196 - 1 = 30,4196 = (25_b_ ⁄ 21_c_)² - 1. Also _vv_ = (19 ⁄ 6)² = 361 ⁄ 36; And _vv_ ⁄ 4_a_ = 361 ⁄ (36 × 64,3596) = 0,15581. Then 30,4196 × 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 Westminster-Bridge._ Altho’ the breadth of the river at Westminster-bridge is 1220 feet; yet, at the time of the greatest fall, there is water thro’ only the thirteen large arches, which amount to 820 feet: to which adding the breadth of the twelve intermediate piers, equal to 174 feet, gives 994 for the breadth of the river at that time: and the velocity of the water just above bridge (from many experiments) is not greater than 2¼ feet _per_ second. Here _b_ = 994; _c_ = 820; _v_ = 2¼; 4_a_ = 64,3596. Now 25_b_ ⁄ 21_c_ = 24850 ⁄ 17220 = 1,443. And 1,443² = 2,082; And 2,082 - 1 = 1,082 = (25_b_ ⁄ 21_c_)² - 1. Also _vv_ = (⁹⁄₄)² = ⁸¹⁄₁₆; And _vv_ ⁄ 48 = 81 ⁄ (16 × 64,3696) = 0,0786. Then 1,082 × 0,0786 = 0,084 feet, the fall sought. Which is about 1 inch; and is about half an inch more than the greatest fall observed by Mr. Labelye. LXIV. _An Account of the Earthquake in the West Parts of_ Cornwall, July _15th 1757. By the Rev._ William Borlase, _M. A. F.R.S. Communicated by the Rev._ Charles Lyttelton, _LL.D. Dean of_ Exeter, _F.R.S._ [Read Jan. 26, 1758.] ON Friday the 15th of July, 1757. a violent shock of an earthquake was felt in the western parts of Cornwall. The thermometer had been higher than usual, and the weather hot, or calm, or both, for eight days before; wind east and north-east. On the 14th in the morning, the wind shifting to the south-west, the weather calm and hazy, there was a shower. The afternoon hazy and fair, wind north-west. The barometer moderately high, but the mercury remarkably variable.