150 Mr. Murphy on the 



or 2 7T *J — ] . root 



o" „ i f, a 



— - 2 6 6 + 2 



= - £l. (c 2e + a e e + b) + ~. tan- ' . + constant, 



and taking these transcendent functions (which enter directly by 

 integration) from to -|- 2 tt S /"^T, the expressions under them do 

 not alter at these limits, and therefore, by the preceding remarks, 

 the value of 



the log. between limits is 2w>/ — 1 



of tan ' w; 



2 7T v/^1 . root = - 2 7T x/^1 . - + 



2 A 



« 2 



7T . 



\/b-i 



or root = — „ 

 2 



£*\/?- 



It need scarcely be added, that this instance is only brought as 

 an illustration of the operation. 



The method, here adopted, for finding the coefficient of the 

 first negative power of x, in any function of x; on which the 

 value of the root of an equation has been made to depend, was 

 chosen, as leading to the preceding remarks on Definite Inte- 

 grals: the application of either Parseval's Method, or of a theorem 

 given in my former paper, on Definite Integrals ; (Vid. Cambridge 

 Transactions, Vol. in. p. 437) to the theorems established in the 

 first 4 Sections, would have sufficed to give other analytical ex- 

 pressions, for the roots of equations, &c. also, in Definite Integrals. 



