Calculus of Operations to Algebraical Expansions. 357 

 which, when x=0, is 



If in the first of these we take D n for <£D, and in the second 

 D n ~ L for <f>T), we obtain 



DVH (when x=0)=n . . . (n-r + l)V n - r t n - r , 



B n ' l (u r (Dt n )^) (when*=0)=rc . . . ( n -r + l)T> n - r t n - r ; 



from which Burmann^s and Lagrange's theorems are immedi- 

 ately deducible. 



One of the most obvions applications of the theorem is to 

 algebraical developments by means of Maclaurin's theorem ; in 

 which it will be found to give a remarkable degree of facility to 

 the determination of the coefficients. 



With the view of obtaining these coefficients in terms of the 

 differences of nothing, I premise the following lemma. 



Lemma. The absolute term of ^A . t n , or its value when t=0, 

 is yjrA . n . For 



^A . f =i|rA(l + A)'. n = (1 + A)', f A . (T =^A . n , 



when / is 0. Or it may be proved by extracting the coefficient 

 of t° in A m ^, and it will be found to be identical with A w n . 



Now in the expansion of <f>x by Maelaurin's theorem, our 

 object is to ascertain the value of [D w . (j>x], that value being the 

 coefficient of #*-*-( 1 .2 . . n). To effect this object, we have 

 [D w <^]=<£v.D\l = coefficient of D° in <£V . D H , or in 

 <£(log(l + A)) . D 7t , which by the lemma is </>(log(l + A)) . W ; 



a formula which expresses the required coefficient in terms of 

 the differences of nothing. The theorem may be thus conve- 

 niently written, 



</># = 0(log(l + A)) . e°* (A operating on 0). 



This theorem may be illustrated by applying it to some of the 



more important algebraical expansions, such as are given in the 



13th chapter of Mr. De Morgan's Differential Calculus, to which 



I shall refer by the number of the section. 



x n 

 1. To expand /(e*—l) in powers of x. Coefficient of r — ^ 



is/A.(T. Thus 



awaw Atnryn+1 



'^ix^^ ix.^i) '"'^" (Sec,6a) 



