igo8.] IN INFINITE SERIES. 113 



In this identity the coefficient of each power of x equals zero. 

 Hence pQ is the root of the two-term equation 



apo + c = o. 



The coefficient of the first power of x equated to zero determines 

 /»! uniquely in terms of p^ ; the coefficient of x^ equated to zero deter- 

 mines p2 uniquely in terms of p^ and p^^ ; in general, the coefficient of 

 x^ equated to zero determines ps uniquely in terms of po,Pi,p2> '"> 

 ps.i. All the successive coefficients of the power series (4) are 

 therefore determined uniquely in terms of p^, any one of the roots 

 of the two-term equation ap^" -\- c^o. 



The power series representing the algebraic functions defined 

 by equations (2) and (3) are determined in precisely the same man- 

 ner. Unfortunately if the coefficients of the power series are deter- 

 mined in this way it is difficult to recognize the law which will 

 enable one to write the general term of the power series, which is 

 necessary for the application of a convergency test. 



When X is made unity, the equations (i), (2) and (3) become 

 the three-term equation 



ay^ + by^ -{- c = o 



and the power series, if convergent when x= i, becomes the solution 

 of this equation. 



If it is known in advance that some one of equations (i), (2), 

 (3) furnishes a power series which is convergent when x=i, the 

 multinomial theorem determines in an elementary and direct manner 

 the coefficients of the power series. 



8. Maclaurin's Series. — The algebraic function y defined by the 

 equation 



ay'^ -j- hy^\x -\- c = o 



can be expanded into a power series in x by means of IMaclaurin's 

 series 



(O) v-V 4.'^^'X4.^^'^4.'^'^' •*'' ^... 



^9^ ^-y^-^dx,''^ dx^'l-2^d7^T^2^7,^ • 



The expansion is identical in form with the expansion obtained 

 by means of the multinomial theorem and consequently has the same 

 disadvantage. 



