354 Astronomical and Nautical Collection!:. 



iutegratetl, we may obtain a converging- series by means oi' the 

 Taylorian theorem ; but we must make the fluxion of the re- 

 fraction constant, and that of the density variable ; so that tlie 



■II u du , d^u ?■' , d'u 7^ , V, ■ 



equation will be u — — r + — +. — +...,ubemg 



^ d;- d/^ 2 dr^ 2.3 



the initial value of «, when r = 0. Now the whole variation, 



of which u is capable, while z decreases from 1 to 0, extends 



from to s; or, since p is very small, from s — ps to s; and 



l+p 



J u • d?< , ^, . , di' r' , 



dr being =: — , we have the equation ps = vr + — — -f ... 

 V dr 2 



_, , , „ , xJ.r — ?<d)t J di) X dx 



But u r= v^ (x' —U-), dv = , and _=: « ; 



t; dr u dr 



J J I • <^y 1 1 , dj,- V dy 



and a.v being = -, and du — —psdz, — - = • ^r-- 



?«z ' dr mpsz az 



4. We must now determine the value of the density z, which, 



when the temperature is uniform, becomes simply = v ; but for 



which we must find some other function of y, including the 



variation of temperature ; and we may adopt, for this purpose, 



the hypothesis lately advanced by Professor Leslie, in the article 



Climate of the Encyclopaedia Britannica, and suppose the density 



to be augmented, by the effect of cold, in the proportion of 



1 to 1 + M j — — z). " being somewhat less than -^^^ ; and since 



the density is as the pressure and the comparative specific 

 gravity conjointlj', we have z — y I 1 +n [" ;j "1 ), — =: 



, , « , z dz zdj/ ndz , , dy y 



1 + nz, d—~ ^^ ndz, and -/ ~ — 



z y y yy zz dz z 



^^^ny^, consequently ^=^L_/jL4.^^.-I^), 

 2' z dr mpsz \z z^ z I 



dij mz dx V nvy nvy , "— w 2nvy 



ydr y ' dr ~ psz psz^ psz ' psz psz^ 



—," which stands in the original paper, in manifest defiance 



psz 



of the first rules of arithmetic : indeed the very first rule of all is 



forgotten, for the paragraphs are mimhered 1 , 2, 3, 4, 6 ... ! But 



