Calculus of Operations to Algebraical Expansions. 361 



or 



The coefficient of # w -f-l . 2 . . ra in e~P x is ±7? w , according as n is 

 even or odd; therefore (1 + A)~ p .0 n , or 



(T - j, A(T+^ +1 > A*0"- ^ + o 1)( / +2) A'(y+ . . 



Similarly, 



KP + I) ; • (j» + n-l) A 4 n_ , ». 



2 



The coefficient of p*M-l . 2 . . n in (fill Vis 



£PQP+n 





(n + l){n + 2)..(n+p)' 

 therefore 



/ a__v ^ A^ + - 



Vlog(l + A)/ ' (n + l)..(n+p) 



Again, since 



we have 

 and also 

 Consequently 



<^=<£(log(l + A)).e% 

 0«# =< p)(log (1 + A)) . 6° A , 

 <^=<£(log (1 + A)) . 0" • €*». 

 ^)(log(l + A)).0^=</>(log(l + A)).O r 



From the nature of the formula which we are exemplifying, it 

 may be expected that its principal applications will be among 

 those theorems which involve operations of differentiation, the 

 variable receiving a particular value after the operations are per- 

 formed. Of these, the principal instances, after Maclaurin's 

 theorem, which has been considered, are the developments which 

 now from Lagrange's and Burmann's theorems, of which I shall 

 take some examples. 



I shall assume as known the coefficients of the expansion of 

 (a + bx + cos 2 + ex 3 -f . . . )~ n in the form 



A„ + B n a? + C n # 2 +E„tf 3 + ..., 

 which are easily found thus : 



Phil. Mag. S. 4. Vol. 6. No. 40. Nov. 1853. ; 2 B 



