128 Mr. Murphy on the 



readily visible: the same Section contains the expression of this 

 law. 



There is besides in this Section, a remarkable expression for 

 the correction to be applied to the root of an equation, when to 

 the equation itself there is added a small term. 



Next in order, the important subject of the Limits of La- 

 grange's series is considered. 



Section V. treats on Definite Integrals : in this Section the root 

 of any proposed equation, any function of the root, &c, are all 

 represented by means of Definite Integrals; and we here see the 

 Analytical use of the multiplicity of the values of certain tran- 

 scendent functions, when the quantity under the sign of such a 

 function should be the same at both the limits of an integral. 



The last Seqtion contains various points of Analysis, connected 

 with the present subject, which could not conveniently be brought 

 into any of the others. 



