156 Algebra. 



Subtract A^^ from the left-hand member and B^^ from the right- 

 hand member of (^i j and we get 



Ax+A,-r^+ . . . -\-Rx"=Bx+BX--\- • • • +RX' (2) 

 Divide both members of (^2 j by x and we get 

 A, + A^jr+ . . -f A,_i-i"-= + RX'"' 



= B^-f B^x'^-f . . . +RX'-'. (3) 

 As before, we have two variables always equal, hence theii 

 limits are equal. 



But as X approaches zero the limit of the right-hand member 

 equals A^ and the limit of the left-hand member equals B^. 

 Hence, by Chapter XI, Art. 7, 



A=B, 

 Repeating the reasoning, we may show successively that 



A^=B„ 



a=b;, 



etc. 



6. The theorem of the last article will enable us to change the 

 form of a function. 



The method of doing this consists in assuming a function of 

 the required form with unknown coefficients and then determin- 

 ing the coefficients so that the function assumed shall be identical 

 with the function proposed. The unknown coefficients are deter- 

 mined b}^ placing the proposed function equal to the assumed 

 function, reducing to the rational integral form, and equating the 

 coefficients of like powers of the variables on the two sides of the 

 equation. 



If the proposed function can be placed in the assumed form it 

 will be found that there are as many independent compatible 

 equations as there are unknown quantities to determine. 



7 . Definition. A function is said to be Developed or Ex- 

 panded when it is expressed in the form of a series, the sum of 

 whose terms when the number of terms of the series is limited, 

 and the limit of the sum when the number of terms is unlimited, 

 equals the given function. 



8. The development of functions is one of the most common 

 applications of the method described in Article 6. The process 



