112 



LAMBERT— SOLUTION OF ALGEBRAIC EQUATIONS [April 25, 



(I) 

 (2) 

 (3) 



11. The Three-term Equation. 



5. In the three-term equation 



ay^^ + ^3'^^ -\- c = o 

 the two terms from which the x is to be omitted can be selected in 

 three different ways. This gives rise to the three equations 



ay^ -\- byKv -{- c = o 

 ay"- + by^ -\- ex = o 

 ay^x -\- by^ -{- c = o 



each one of which defines 3; as an algebraic function of x. 



6. Values of y expressed as power series in x may be found from 

 each one of these three equations by any one of the following three 

 methods, which, however, are essentially the same. 



7. The Multinomial Theorem. — Assume that the power series 

 for y is 



(4) 3' = /'O + Pv'^' + P2^'- + P3-V^ + Pi-V"^ H • 



The multinomial theorem asserts that the coefficient of x^ in the 

 expansion of y" is 



\ n \ n I . . . /7 I ^tJ ^1 ^2 



(5) 



2 



q^-qi'-q-z'- ■••(!. 





provided 



(6) gi + 2go + 3(?3H sqs = r 



(7) ^o + (?i + (?2H qB = n. 



The expansion of 3;^ is obtained in like manner. 



Assuming that the power series (4) represents the algebraic 

 function defined by equation (i), the substitution of the expansions 

 of y" and y^ in equation ( i ) must give an identity. This identity is 



o^apiA-anf-^p^ 

 (8) 



,^-(^^„_^.. 



i{n-\\n-2) 



I • 2 





Jran{n-i)pi-^P,p^ 



X^-\-' 



