22 Mr green, ON THE LAWS OF THE EQUILIBRIUM OF FLUIDS. 



by which n\ is affected. But by omitting these superfluous accents, 

 we shall have to calculate the value of the quantity 



I-n 

 2 , 



fj.dfi. (^) . (r' - 2 rr'/m + r") 

 where 



,. . i.i — 1 . i.i-i,i-2.i-3 ; , - 



^'^ = ''-2:2^:1'' + 2.4.2i-1.2i-3 -^ -^^- 



The method which first presents itself for determining the value of 



l-n 



the integral in question, is to expand the quantity {r^ — 2rr'/u. + r'^) ^ by 

 means of the Binomial Theorem, to replace the various powers of m by 

 their values in functions similar to (i) and afterwards to effect the in- 

 tegrations by the formulee contained in the third Book of the Mec. Cel. 

 For this purpose we have the general equation 



.-s i ... , i.i — 1 ,. „. , i.i—l.i — 2.i — 3,. ,, 



^^^ '^ =^^)+ 2:271:1 (^-^^-^2X2I33:27::5(*-*) 



i.i-l.i-2.i-3.i-4!.i-5 . 

 2.i.6.2i-5.2i-7.2i-9 ^' '") + ^^' 



To remove all doubt of the correctness of this equation, we may 

 multiply each of its sides by (i, and reduce the products on the right 

 by means of the relation 



which it is very easy to prove exists between functions of the form (?). 

 In this way it will be seen that if the equation (9) holds good for any 

 power fx' it will do so likewise for the following power ^'+^ and as it 

 is evidently correct when i='l, it is therefore necessarily so, whatever 

 whole number i may represent. 



Now by means of the Binomial Theorem, we obtain when r^r' 



= 2, 



r"'-K(r'-2rr'^ + r"y = (i-2m J + ^,) 

 y>n—l.n + l.n-\-3 n + 2s — 3 



l—n 

 3 



2s 





If now we conceive the quantity (2ju- rj to be expanded by 



