THE EQUILIBRIUM OF FLUIDS. 141 



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

 may multiply each of its sides by /u,, 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 (i). In this way it will be seen that if the equation (9) 

 holds good for any power p* it will do so likewise for the follow- 

 ing power //*"*, and as it is evidently correct when i = 1, it is 

 therefore necessarily so, whatever whole number i may represent. 



Now by means of the Binomial Theorem, we obtain when 

 r<r' 



/ r r z \ 



= ^ _ 2 r + r \ 

 V r r J 



2, 



If now we conceive the quantity f2//,-, ^J to be expanded 

 by the same theorem, it is easy to perceive that the term having 

 for factor is 



, r \ 

 (-, j 



i+2 ^ 



i+2t , r i+ztf 



2.4... 



n^ 

 ' 





2.4... 

 &c ......................... &c ...................... &c ............. 



/y.y+2*- < 

 and therefore the coefficient of (-,1 in the expansion of the 



function 



