26 Mr green, on THE LAWS OF THE EQUILIBRIUM OF FLUIDS. 



If now we restore to n. the accents with which it was originally 

 affected, and multiply the resulting quantity by r'""\ we shall have when 

 r<r'' 



(10) /Ad^\(i) if-^rr't^, + ■' =/»-y_;</M'i ii) (l - UJ-. + ^-.) 



_ ,j_„ 1.2.3 i w-l.w + 1 ?^ + 2^^ + 2^-3 



~ '1.3.5 2«-l 3 . 5 2e + 2#'+l 





;« — 2.W « + 2#'-4 /r 



2.4 2jf' 



9 



j + 21' 



and in order to deduce the value of the same integral when /•' /. r, we 

 shall only have to change r into /, and reciprocally, in the formula 

 just given. 



We may now readily obtain the value of Vi^ by means of the 

 formula (8). For the density corresponding thereto being 



:/;«/•'+=' (l-r"7, ■ 



it follows from what has been observed in the former part of the 

 present article, that ^'®r'^^' may always be reduced to a rational and 

 entire function of icf, y, %' the rectangular co-ordinates of the element 

 dv, and therefore the density in question will admit of being expanded 

 in a series of the entire powers of x, y', %' and the various products of 

 these powers. Hence (Art. 1.) F/'' admits of a similar expansion in 

 entire powers, he. of x, y, z the rectangular co-ordinates of the point p, 

 and by following the methods before exposed Art. 1 and 2, we readily 

 get 



^t ^-^J J r' ui y>. , ) .^ 3.5 2t + 2t' + l 



n-2.n.n + 2 » + 2#' — 4 /r\'+"' 



X 



2 .4. 6 2t' 



-4 /r^y^-"" 



" V'j ' 



and thence we have ultimately, 



(,ii; rt Airj, ^33 2i + 2t' + 1 2.4 2/' 



