126 Mr. Murphy on the 



I have supposed the given equation to be such as to contain 

 neither negative nor fractional powers of the unknown quantity x ; 

 if such should enter any proposed equation as <p(x) = 0, we need 

 only put x = a + z, and consider s as the unknown quantity, since 

 <f>(a + z) may always be expanded according to the positive and 

 integer powers of %, when no particular value is assigned to a, 

 this is therefore to be understood, unless where the contrary is 

 expressed. 



Suppose the root of an equation (j>(x) = is sought, the fol- 

 lowing simple rule which I have proved and applied in Section 1. 

 will give it with great facility. 



" Divide the given equation by x, take the Nap. log. of the 

 quotient by means of the formula 



1.1 +* = *-- + -, &c. 



take the coefficient of the first negative power of x in this loga- 

 rithmic expansion : this, with its sign changed, is the root of the 

 proposed equation." 



If the proposed equation were of n dimensions, it has n roots, 

 and it is natural to enquire which root is given by the preceding 

 method. I have in the same Section shewn that it analytically 

 gives a result which comprehends all the roots, but that arith- 

 metically it gives the least root. 



If, instead of the root of an equation, any function f{x) of the 

 root should be required, there is given in Section (2) for this 

 purpose, a rule nearly as simple as the above; namely, 



" Take the same Naperian log. as before. Multiply it by 

 the derived function f(x), the coefficient of the first negative 



