356 Mr. C. J. llargreave's Applications of the 



If the coefficient A be zero, or if x— a, or (x— a) n be a factor 

 of the original expression, the case involves some considerations 

 which render it necessary to proceed with caution. Let us con- 

 sider the equation 



D w (>-a)-V) (when x=a) = V~ l D n u= ^y J) n+1 u, 



which, as it is obtained by the formula, might appear to be true 

 universally. But we have 



- (x-a) n+l ' 



which becomes infinite when x = a. 



If, however, u itself should contain a, and have n + 1 roots 



equal to a, then each term assumes the form -^ ; and, obtaining 



these singular values by differentiation of the numerators and 

 denominators, it becomes 



* W V"2 + ~273 27374" + " J ' 



which is j—y D n+1 M, being the result obtained by the formula. 



In the same way 



D n ((x—a) m ii) (when a?=a)= ? — ; — - 



V' y ' v y (n+l)(n + 2) . . .(n + m) 



may be verified, if & has m-f 7i roots equal to a. We are not, 

 however, ordinarily to regard u as containing a. 



4. Let ^jrx=a -\-a^x + . . -f««# n ; then, since 



we have 



0D{(« o +a l *+...a n ^) X *}=^^ 



when a?=a, a root of the expression tyx. 



5. Let m be a function of x such that x-t-u (or /) is expandible 

 in positive powers of x, then 



<P'D{u r t H )=<t)T){x r t n - r ) ) 



which, when a?=0, is ^D(/»- r ). 

 Again, we have 



<^D(w r (D/ n )) = </)D(^-''(D/ n ))=^;</)D(^(Dr-')) 



