62 DIFFERENTIAL AND INTEGRAL CALCULUS. 



or!f 



(We have assumed 7 to be positive, if it be negative we 



ought to take the other sign in the ambiguity). 

 Putting a in the general formula, we have 



... 



*-J2T* 



or, replacing h by a?, 



/(*) =/(o) + xf (0) + f (o) +. . .+pT(fe), 



6 denoting some positive proper fraction ; and all the 

 functions /(#), f'(x)...f n (x) being finite between the 

 limits and x. Thus, if f(x) be log a?, we cannot obtain 

 an expansion since /(a?), /'(a?)... all become infinite when 



ce = 0. So also if /(a?) be e~i , e~.... 



If y(aj) or ?/ be sin" 1 (a;), "we have seen that 



hence, putting x = in this equation 



from which we can find /" (0), /'" (0)... after finding /((>) 

 and/' (0). But/(0) = 0, /' (0) = 1, whence 



/"(0) = 0, /'-(0)=0,/"(0) = 0, ..., 

 and /'" (0) = 1, f (0) = 3\ f" (0) = 3 2 .5", ..., 



or 



Li Li LL 



la; 3 1.3 a; 5 1.3.5 a; 7 



