Ross — Verb- Functions. 



45 



that of § 13, with the sole difference that the dividend has only a 

 single term /3. As the process can always be applied, the value of 

 [0]"^ can always be obtained, no matter what its degree, and always 

 by the same method ; but that value will in general be an infinite 

 series. 



If the first term of the divisor be y8", the first term of the quotient 



must be yS", since [^"]~^ = ^ Hence, in order to obtain the first 



subtrahend, we must expand the-^'* power of by the binomial or 



multinomial theorem ; and similar expansions must, in general, be used 

 for each successive subtrahend. But it will be seen later that the 

 quotient can easily be written out by means of a general rule. As (B 

 has no quantitative value, the question of convergency of series does 

 not generally arise in connexion with the operative expressions ; and 

 their accuracy can be established as an identity by reversing the pro- 

 cess of division and applying the quotient to the divisor, or, what is 

 the same thing in this case, the divisor to the quotient. The process 

 can best be studied by the aid of a few examples. 



(1) Find a root of the equation 



+ ax + h = 0. 



Then 



X + a-'x' = - and x = + a-'/3'y (- a~'h). 



- a' 13' - ba--(3' - \0a-^(3'^ - R 



5a ^13'+ I0a-'I3''+ R 

 ba--(3' + 45«-3/?^^ + R 



- S5a-'/3''-R; 



.-. x = - a-'h + a-'P - ba-'^'h^ + Zba-H'^ - R. 



This has only one value. 



