140 THE LAWS OF 



dv. If therefore in the expression immediately preceding, we 

 make 



Y' (i} =f t ' (i) and f (g z ) = g n ~ l = (ff*)~** , 



and substitute the value of the integral thus obtained for its 

 equal in V t (i) there will arise 



V t u= 27T/f I'l'l'"^ 1 J V^W (1 -ryj 1 dn\ (i) 



where f t (i) is deduced from f t ' (i} by changing 0', OT' into 6, OT, and 

 (i), for abridgement, is written in the place of the function 



n _ fr-ft~ * ' i-s i.i l.i2. i- 3 ,<_ 4 



As the integral relative to /// x which enters into the expression 

 on the right side of the equation (8) is a definite one, and depends 

 therefore on the two extreme values of // x only, it is evident that 

 in the determination of this integral, it is altogether useless to 

 retain the accents by which p l is affected. But by omitting 

 these superfluous accents, we shall have to calculate the value of 

 the quantity 



I dp (i) . (r z 2rr'fjL + r' 2 ) * ; 



J-4 



where 



i.i-l , , . i.il.i-V.i- 3 



{ 



The method which first presents itself for determining the 

 value of the integral in question, is to expand the quantity 



l-n 



( r 2 _ 2rr'fj, + r' a ) 2 by means of the Binomial Theorem, to replace 

 the various powers of p by their values in functions similar to 

 (i) and afterwards to effect the integrations by the formulas 

 contained in the third Book of the Mec. CeL For this purpose 

 we. have the general equation 







