278 PHILOSOPHICAL TRANSACTIONS. [aNNO 1758. 



this globe, not without giving some attention to the peculiarities of each climate ; 

 and I can fairly pronounce that I never felt such heats any where as in Georgia. 

 1 know experiments on this subject are extremely liable to error ; but I presume 

 I cannot now be mistaken, either in the goodness of the instrument, or in the 

 feimess of the trials, which I have repeatedly made with it. This same thermo- 

 meter I have had thrice in the equatorial parts of Africa ; as often at Jamaica, 

 and the West India islands ; and on examination of my journals, I do not find 

 that the quicksilver ever rose in those parts above the 87 th degree, and to that 

 but seldom : its general station was between the 79th and 86th degree ; and yet 

 I think I have felt those degrees, with a moist air, more disagreeable than what 

 I now feel. 



In my account of the late expedition to the north-west, I have observed that 

 all the changes and variety of weather that happen in the temperate zone through«- 

 out the year, may be experienced at the Hudson's Bay settlements in 24 hours. 

 But I may now extend this observation ; for in my cellar the thermometer stands 

 at 81, in the next story at 102, and in the upper one at 105 ; and yet these 

 heats, violent as they are, would be tolerable but for the sudden changes that 

 succeed them. On the 10th of December last the mercury was at 86; on the 

 11th it was so low as 38 of the same instrument. What havock must this make 

 with a European constitution ? Yet but few people die here out of the onlinary 

 course; though indeed one can scarcely call it living, merely to breathe, and 

 trail about a vigorless body ; yet such is generally our condition from the middle 

 of June to the middle of September. 



CIII. The Invention of a General Method for determining the Sum of every 

 id, 3d, 4th, or 5th, &c. Term of a Series, taken in order', the Sum of the 

 whole Series being known. By Tho. Simpson, F. R. S. p. 757* 

 As the doctrine of series is of very great use in the higher branches of the ma- 

 thematics, and their application to nature, every attempt tending to extend that 

 doctrine may justly merit some degree of regard. The subject oi the present 

 paper will be found an improvement of some consequence in that part of science. 

 And how far the business of finding fluents may, in some cases, be facilitated 

 by it, will appear from the examples subjoined, in illustration of the general 

 method here delivered. 



The series propounded, whose sum (s) is supposed to be given (either in alge- 

 braic terms, or by the measures of angles and ratios, &c.) is here represented by 

 a ■\- bx -^ cx^ -^ dx^ + ex'*^ &c. and Mr. S. first gives the solution of that case, 

 where every 3d term is required to be taken, or where the series to be summed 

 is a ^ da^ -{- gx^ + kx'^ &c. By means of which, the general metliod of pro- 

 ceeding, and the solution of ^stry other case, will appear evident. 



