VOL. LXXVII.] PHILOSOPHICAL TRANSACTIONS. IQl 



and roads, which looked like smoke, and had the appearance at a distance as if 

 many villages had been on fire all over the country. The water too was raised 

 from the surface of the river, and carried quite away by the violence of the 

 hurricane, forming small clouds in the air, which we traced to a great distance. 

 The great violence of this tempest lasted but a few hours, and at night it fell 

 calm. The barometer was little affected, and stood at '294- inches. The wind 

 was nearly west, veering sometimes a little to the north. From this time we had a 

 course of very fine weather, the wind constantly in the west points, till the time 

 of the earthquake (which happened on what is called the lastof the dog days), when 

 it changed to the south-east, and brought us 5 of the worst days I ever re- 

 member to have seen at that season ; it rained almost incessantly, with a cold 

 easterly wind, and the sun did not once appear till the morning of Wednesday 

 the 16th, after which we had again a course of fine weather. I examined the 

 barometer at the time of the earthquake, but did not find that it had been sen- 

 sibly affected. It rose a little on that morning ; but this I imputed to the wind 

 having changed into the east. 



XIV^. On Jinding the Values of Algebraic Quantities by Converging Serieses, 

 and Demonstrating and Extending Propositions given by Pappus and others. 

 By Edw. rearing, F.R.S. p. 71. 



Suppose the roots of the equation a;* + 1 = O to be given, where h denotes 

 any whole number or fraction ; to find the roots or values of any given alge- 

 braical qnantity, by converging infinite serieses. 



1. Let the algebraical quantity be ^(4: a), then the roots of the algebraical 



* i i 



quantity will be A« X (a + A ^ — l), a«" X (P + /* \/ — J), A" X (•y-f-vv' — l), 



&c. where « -|- a \/ — 1, |3 -|- i<* >/ — 1, y + v\^—ly &c. are the roots of the 



equati n a^ + 1 = O. ft will be -j- 1 if it was — a, and — 1 if -|- a. 



2. Let the given algebraical quantity be [/{± v'(i:^)± v^(±b)± ;J/± 



c + &c.), and « + x \/— 1, «' -|- V -/ — 1, «" + x'V — J, &c. and r -f- A 



v/ — 1 , be respectively one of the roots of the equations of :f 1 = O, a?™ ip 1 =0, 



1 I 



a* + 1 = O, &c. and x ^I 1 = O. Substitute + p = + a« a + b« a + c« «" 



I I i^ 



+ &c. and + a = ± a^x + b^ a' + c^x'' ± &c. In the first place let p be 



I I 



greater a, and ± p be -j- p, then will (p ±a v^ — l)*- = (p*- . ^-"IT ^ 



-^ - &c. = + l) + (^ -^ 1 i 



1 - 3r 1 - 4r Q^ 



