400 



VHILOSOPHICAL TRANSACTIONS. 



[anno 175s. 



Which expression is equivalent to the sum of the following series ; for 



it 



~-j^ = 2t-\-2e+ ie + 2/' + 2/», &c. 



2t 



21 



z(——y 



^1 — tt' 



2t 



128zt' + 8Q6zt^, &c. 



512«<», &c. 



And, by making an equation between 2x, the co-efficient of t* in the first 

 found value of 2a, and 2 -f 8t, the sum of the co-efficients of l^ in the latter, 

 gives 2ar = 2 -|- 6x; hence — -j- = ar. 



And by proceeding in the same manner with the co-efficients of /*, f, t", &c. 

 and supplying the places of x, y, z, &c. as they arise, by the numbers so found. 



u (— — )9 = 



we shall have 



23,= 2 -V+ 323,; 

 2^=2-^-|-^-f 128z; 



hence y = -)- 4-. 

 hence z = — \. 



2M = 2-V-j-*-|?- 52^-1-5 12m; hence m = -|- i- 



Therefore we may conclude that t 



i<' + 



.f -j- ^<«, &c. = a. 



When the arc is just 45 degrees, then < = 1 , and the series becomes t — -r + 

 4- — -f -|- -^, &c. which converges exceedingly slow ; but, by the assistance of a 

 method, given in the appendix to M. de Moivre's Miscellanea Analytica, it may 

 be transformed to another, converging quicker ; which method is applied to this 

 very series, in folio 362 of the Mathematical Repository, vol. i. 



XLI. A Letter from John Lining, M.D. of Charlestown, South-Carolina, con- 

 cerning the Quantity qf Rain fallen there from Jan. 1738, to Dec. 1752. 

 p. 284. 

 As there are thunder-gusts here in the Months, The Means 



&c. 



hot months, in which a vast quantity of j^^ ' ^ 



rain falls, the depth of the rain in these 

 months is greatly increased; for there 

 is very little rain, excepting in thunder- 

 showers. Thus, on June 30, 1750, in 

 a thunder-storm, there fell, in 24 hours, 

 5.335 inches of rain. On September 

 16, 1751, there fell, in 24 hours (but the greatest part in 6 hours) 9.955 

 inches of rain. On Sept. 15, 1752, during the time of the most violent hurri- 

 cane that was ever felt in this town, the depth of rain which fell, was only 3.740 



Feb. . . 

 March 

 April . 

 May . . 

 June . . 

 July . . 

 Aug. . 

 Sept. . . 



326' 

 3.389 

 3 024 

 1.721 

 3.655 

 5.000 

 6.149 

 7..>30 

 6'. 343 



Months, The Means. 

 &c. 



Oct 3.049 



Nov 2.229 



Dec 3.684 



Spring 8.<68 



Summer . . 14.804 



Autumn .. 16.913 



Winter • •. . H.340 



Total Depth 48.023 



