586 PHILOSOPHICAL TRANSACTIONS. [ANNO 1784. 



and evaporating from the outside, the remainder in the vessel becomes cold 

 enough to freeze ; the warmth of the earth being at the same time intercepted 

 by the vessels being placed on bodies little disposed to conduct heat.* If ice is 

 thus producible in a climate where natural ice is never seen, we need not wonder 

 that congelation should take place where the same principle operates amidst 

 actual ice. 



It has been observed above, that the heat emitted by the congealing vapour 

 probably unites with and liquefies contiguous portions of ice ; but whether the 

 whole, either of the heat so emitted, or of that originally introduced into the 

 funnel, is thus taken up ; how often it may unite with other portions of ice, and 

 be driven out from other new congelations ; whether there exists any difference 

 in its chemical affinity or elective attraction to water in different states and the 

 contiguous bodies ; whether part of it may not ultimately escape, without per- 

 forming the office expected from it on the ice ; and to what distance from the 

 evaporating surface the refrigerating effect of the evaporation may extend ; must 

 be left for further experiments to determine. 



XXVlll. On the Summation of Series, tvhose General Term is a Determinate 



Function of z the Distance from the first Term of the Series. By Edw. 



Waring, M. D., F. R. S. p. 385. 



Prob. The sum s being given, to find a series of which it is the sum. 



1. Reduce the sum s into a converging series, proceeding according to the di- 

 mensions of any small quantities, and it is done. For example : let any alge- 

 braical function s of an unknown or small quantity x be assumed, reduce it into 

 a converging series proceeding according to the dimensions of x, and there re- 

 sults a series whose sum is s. 2. Let a, b, c, &c. be algebraical functions of x; 

 reduce the /a x, /bx, jcx, &c. into a converging series, proceeding according to 

 the dimensions of x, and the problem is done. 



It is always necessary to find the values of x, between which the above-men- 

 tioned serieses converge. Reduce the algebraical function s in the first example, 

 and the algebraical functions a, b, c, &c. in the 2d into their lowest terms ; and 

 in such a manner that the quantities contained in the numerator and denominator 

 may have no denominator : make the denominator in the first example, and the 

 denominator in the 2d, and every distinct irrational quantity contained in them, 

 respectively =0; and also every distinct irrational quantity contained in the mu 

 merators = 0. Suppose * the least root, affirmative or negative, (but not = O) 

 of the above-mentioned resulting equations ; then a series ascending according 

 to the dimensions of x will always converge, if the value of x be contained be- 



* See a description of this process in the Phil. Trans, vol. 65, p. 253. — Orig. 



