358 Mr. C. J. Hargreave's Applications of the 



a* 



2. To expand /][€*) ; Herschel's theorem. Coefficient of 



is /(1 + A).0 n . (Sec. 160.) 



3. To expand - — =-. Coefficient of -= — s is 



r e*— 1 1 .2. .n 



l£S(l_tA)o» ) orO«-^ + ^*-... + ^ 

 A 2 3 ~~ n + 1 



which is therefore the general expression for Bernoulli's numbers. 



1 x n 1 



4. To expand .. . Coefficient of = — ~ is ^ — -r-0 n , which 



is l(o«-^ + ^-t™L + .. + *^). (Sec. 17.) 



X 



The expansion of — — =- may be thus obtained : 



[D n ( ? ^)] = [ 6 ^ II I) n .^] = [2D M .^] = coefficient of 

 Din£D n . Now 



^(T.D + Air. y + AV. D Py > + ..(,ec.41); 



and the required coefficient is (T— g A0 n + 3 A 2 n - . . as before. 



In like manner to expand ( _ ..- ) , we have 



[D n 7^iy2 J = [X{W n ).x*] = twice the coefficient of D 2 in S(2D»). 

 Now 



2(2D » )=0 ».M F i) +AV .5tti 



D(D-l)(D-2)(D-8) 

 + Z ^ * 2.3.4 + •••' 



and the required coefficient is 



A 3 n +...)- 



Proceeding in a similar way to expand ( x_i ) > we sna ^ 

 require to find (1 . 2 . 3 . . p) x coefficient of D p in 2 p D n . This 



