126 LAMBERT— SOLUTION OF ALGEBRAIC EQUATIONS [April 25, 



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

 in ten different ways. This gives rise to the ten equations : 



(44) 03;" -)- by^'x -f- cyKv -|- dy"^x -\- l = o 



(45 ) ay'^ -|- by^^ + cy^x -{- dy^'Kv -\- Ix = o 



(46) ay"^ -f by^x + cy' H~ dy'^x + ^-v = o(jl 



(47) aj/" -f- by^'x -\- cyKv -\- dy'"^ -)- Ix = o 



(48) ay"x -\- by^ -}- cy^ -\- dy"Kv -\- Ix = o 



(49) ay^'x -\- by^ -\- cyKv -\- dy"'' -{- Ix = o 



(50) ay"x -\- by^'x + cy^ -\- dy"^ -\- Ix = o 



(51) ay'\v + ^3'''" + cy ^x -{- dy'Kv -f / = o 



( 52 ) ay'Kv -\- by^'x -\- cy^ -\- dy^Kv -{- l = o 



( 53 ) ay"x -\- by^x -f cy Kv -\~ dy'^ -|- / = o 



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

 tion of X which may be expanded into a power series by any one of 

 the methods of articles 7, 8, 9. 



25. The terms of the power series expressing the value of the 

 algebraic function defined by equation (44), using the symbol (14) 

 and placing 3'o^ ( — l/ay^^\ when x is made unity, may be arranged 

 as follows : 



