Resolution of Algebraical Equations. 129 



SECTION I. 



To find the root of any equation <j>(x) = 0, which contains only 

 positive and integer powers of x. 



Divide the equation by x, take the Nap. log. of the quotient ; 

 the coefficient of the first negative power of x, with its sign 

 changed, is the root of the equation. 



For example, in the quadratic equation 



x~ + ax + b = 0, 



divide by x, and take the log. of the quotient, 



K 



/ b\ ( x + 'x\ 

 i. 1. la + x + -), or 1. a + 1. \1 H J , 



l. e. 



by the preceding rule the root should be therefore the coefficient 



of - in 



x 



>2» / b\* ( b\ 3 



X + - [X + - ) [X + - ) 



X , v x) , V xl , „ 



a * a- J a 



and selecting the coefficient of - which enters only in the 1 st , 

 3 rd , 5 th , &c. terms, we get for the required root 



U> b 2 4 V 6.5 ^ 8.7.6 V .\ 



~ Q + « 3 + 2"« 5 + 2.3"« 7 + 2.3.4-« 9 ' C T 



as it evidently is, since this series represents the expansion of 



a ,\i 



Vol. IV. Part I. R 



