118 LAMBERT— SOLUTION OF ALGEBRAIC EQUATIONS [April 25, 



(b) The fraction to the right of the sign of inequahty is obtained 

 from the fraction to the left by replacing each coefficient by its 

 exponent. 



(c) The sign of inequality is < when the term containing x is 

 between the other two terms ; if the term containing x is an end term 

 the sign of inequality is >. 



14. The following table exhibits the convergency conditions for 

 the series obtained from equations (i), (2), (3) and the number of 

 roots of the three-term equation 



ay" -f by^ -\- c^o 



furnished by each one of these series. 



/;" n" 



( ' ) ^/' + ^J'"''' + ^ = ° " «V-"^ = k'Xn-ky'^' ' 



(20) (2) af + bf + ex = n-k r ^» ^ ;/" 



(3) afx + bf + ^ = o k 1 ~^^' ^ J^\n-I^^^ • 



The roots of the three-term equation can always be expressed 

 in infinite series. 



III. The Four-term Equation. 



15. In the four-term equation 



ay" -)- by^ -[- cy ^ -j- rf = o 



the two terms from which the factor x is to be omitted can be 

 selected in six different ways. This gives rise to the six equations : 



(21) ay" -j- by'^x -^ cy^x -|- (/ = o 



(22) ay^ 4" ^3''^' + O''-"^" + ^^■^' "=" o 



(23) ay"x -f- ^3''^ + O''-^" -j- (/ = o 



(24) ay'^x -{- by'' -\- cy' -\- dx = o 



( 25 ) ay^x -\- by'-'x -\- cy'' -\- d = o 



(26) oy" -\- by'^'x -)- cy'- -\- dx = o 



Each one of these six equations defines y as an algebraic func- 

 tion of X. The y of equation (21) may be expanded into a power 

 series in x by any one of the three methods of articles 7, 8, 9. 

 Using the symbol (14) and denoting ( — d/ay^" by y^, this power 

 series, when x is made unity, becomes 



