144 Mr. Murphy on the 



1. e. 



it^a 1 ' 8 •"?*" V^'V t°' subject to the two conditions 

 1 1.2...i.xl.2 ...4 1S &c. J 



&„ + *!+ i„, &C. = W, 



J -6 l -2ft 2 , &c. = l. 



When to an equation F(a;) = we add a very small term <?.<£(#), 

 what is the corresponding correction to be applied to the root? 



The root of F(x)=0, 



1 F (x) 



is the coefficient of - in - 1. — ^ , 

 x x 



and the root of F (x) + e<j> (x) = 



is the coefficient of - in - 1. — — + -^-\ ; 

 therefore the increment of the root 



= coefficient of - in — 1. ( — + ) + 1. — 



X \ X X I X 



= coefficient of - in 1. jl + ■£, . [ 



= coefficient of - in ~, . 



x i< (x) 



when <?$(#) is supposed to be very small. 



Cor. When the proposed equation is as=o +/(*), which by 

 adding an additional small term, becomes 



z = a+f(z)+e<p{z), 

 the correction to be applied to the root is 



