550 LAPLACE. 



Similarly 



Thus we have approximately 







"7r-,+ 



,-r+.^^«+r+| JI9rrA 1 2ww 6m^ G^l' 



Now suppose that the values of m and w are those which we 

 have already assigned as corresponding to the greatest term of 

 the expansion of (i? + ^)^ then 



171 — 2 n + z 



thus we have approximately 



m-r n+r^ U + ^^ • 



t ^ ^^ \ ran) , 



Therefore finally we have approximately for the r*^ term after 

 the greatest 



tji^irmri) \ mn 2mn 6m^ 6n^ 



We shall obtain the approximate value of the r"^ term before 

 the greatest by changing the sign of r in the above expression ; 

 by adding the values of the two terms we have 



2^/ll .El 



i^i^irmn) 



If we take the sum of the values of this expression from r = 

 to r = r, we obtain approximately the sum of twice the greatest 

 term of a certain binomial expansion together with the r terms 

 which precede and the r terms which follow the greatest term ; 

 subtract the greatest term, and we have the approximate value of 

 the sum of 2^ + 1 terms of a binomial expansion which include 

 the greatest term as their middle term. 



Now by Euler's theorem, given in Art. 33i, 





