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



we have for the determination of V n the equation 



A n 0\V n + A H ~ 1 n .V n _ 1 + ... + A0^ 1 + n V o =J^ ( S ec. 61.), 



71 ~\~ J. 



from which the values of V„ can be found by giving to n suc- 

 cessive numerical values. 



In like manner, since the coefficient of x n -*■ 1 . 2 . . n in 

 /6*-ly. A"<y +W 



(— J 1S (»+l)(.i + 2).. ■(»+!>)' WC haVe 



\log(l + A)/ U -(» + l)(n + 2)... (»+/>)' 



and, the expansion of L — 77-37x1 ) beingV + Y l A + V 2 A 2 + . . ., 

 we determine V- from 



<w+* 



A n n .V n +A n " , n .V n _ 1 + ...= 



(*+l). .(*+/*) 



Since the coefficient of x n -t- 1 . 2 ... n in log (1 —x) is 



-(l.2.8..(n-l)), 

 we have for the determination of V n in 



log (1- log (1 + A))=V 1 A + V 9 A 2 + . . . V n A« + . . . 

 the equation 

 A^.V^+A^^.V^^.-.+AO"^^-^^.^.^-!)). 



In like manner, if 



(l-log(l+A))- , =V + V 1 A + V 2 A 2 +..., 

 we have 



A"0\ V n + A"" 1 *)* *&«+» • • +0«V =1 . 2 . 3 . . . n; 



and so on for other developments. 

 Generally, if 



£(log(l + A)) = V +V 1 A + V 2 A*+ ..., 



we determine the coefficients V , \ l . . by the equation 



A n n . V w + A^'O* . V;., + . . + A0\ V, + 0\ V tt = [f»>a?] . 



Various properties of the differences of nothing may be ob- 

 tained very simply by this method. Thus, since the coefficient 

 of x n -5-1 . 2 . . n in x m is 0, save when n=m, in which case it is 

 1 . 2 . . m, we perceive that (log (1 + A))™ . n is 0, except when 

 n= m, in which case it is 1 . 2 . . m. (Sec. 166.) 



The coefficient of #"-5-1 . 2 . . n in xe* is n; therefore 



(1 + A)log(l+A).0 n , 



