VOL. LXX.] PHILOSOPHICAL TRANSACTIONS. 625 



or involve in it an infinite series. According!)-, the conclusions obtained in this 

 paper are as follow: If i?r is less than Vt9^> but greater than its half, and e be 

 put, as before, = -^r, and zz = -^^q^ — \rr, it is shown in the present paper, 

 that the root of the equation x^ — qx = r will be equal to the mixed expression 

 ^v(e + z) + s3/(e — z) + 4^e X the infinite series — 4- >-*-^" + J^^i!! 



+ &c. 



As to the 2d branch of the 16. case of the equation x^ — qx =z r, or that in 

 which \rr is less than half -^'-q^, I do not know, says Mr. M. any method of 

 extending Cardan's rule to it. But I have been informed by my learned and in- 

 genious friend Dr. Charles Hutton, professor of mathematics in the Royal Aca- 

 demy at Woolwich, that he has discovered such a method: and I hope he will 

 soon communicate it to this learned Society. 



VI. An Account of a Method for the safe Removal of Ships that have been driven 

 on Shore, and damaged in their Bottoms, to Places, however distant, for 

 repairing them. By Mr. IVm. Barnard, Sliipbuilder, Grove-street, Deptford. 

 p. 100. 



On January the 1st, 1779, in a most dreadful storm, the York East India- 

 man, of 800 tons, homeward bound, with a pepper cargo, parted her cables in 

 Margate Roads, and was driven on shore, within 100 feet of the head,' and 30 

 feet of the side, of Margate Pier, then drawing 11 feet 6 inches water, the flow 

 of a good spring tide being only 14 feet at that place. On the 3d of the same 

 month Mr. B. went down, as a shipbuilder, to assist as much as lay in his power 

 to save the ship. He found her perfectly upright, and her shere, or side appear- 

 ance, the same as when first built, but sunk to the 1 2 feet water mark, fore and 

 aft, in a bed of chalk mixed with a stifle blue clay, exactly the shape of her body 

 below that draft of water; and from the rudder being torn from her as she 

 struck coming on shore, and the violent agitation of the sea after being there, 

 her stern was so greatly injured as to admit free access to it, which filled her for 

 4 days equal to the flow of the tide. Having fully informed himself of her situa- 

 tion and the flow of spring tides, and being clearly of opinion she might be again 

 got ofi^, he recommended, as the first necessary step, the immediate discharge 

 of the cargo; and, in the progress of that business, he found the tide always 

 flowed to the same height on the ship; and when the cargo was half discharged, 

 and he knew the remaining part should not make her draw more than 18 feet 

 water, and while he was observing the water at 22^ feet by the ship's marks, she 

 instantly lifted to 17 feet 8 inches, the water and air being before excluded by 

 her pressure on the clay, and the atmosphere acting on her upper part equal to 

 600 tons, wiiich is the weight of water displaced at the difference of those two 

 draughts of water. 



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