148 Mr. Murphy on the 



the common difference of that progression is the quantity by 

 which they alter through one circulation ; which is therefore 



2tt for sin -1 (x) ; ■* for tan _1 .r, and 2?r sj — 1 for log (x) : 



if then at the limits of an integral, the quantity under any of 

 these signs should be the same ; and if x has been supposed to 

 have passed through one circulation, from its leaving a certain 

 limit, until it has returned to the same, then for the correspond- 

 ing part of the definite integral, we evidently must not put 0, 

 but the common difference of the above-named progression. 



Thus, in finding the area of a circle f ■' , taken from 



•\/l - x : 



v = 1 until x = 1 after one circulation, the value is sin -1 (1) - 

 sin _1 (l)» but the former must be manifestly understood to exceed 

 the latter by 2tt. 



If this distinction be attended to, there will arise no difficulty, 

 in calculating the values of terms of this nature when they enter 

 definite integrals, as they frequently do ; we have also here the 

 advantage of seeing the analytical use of the multiplicity of the 

 values of these kind of functions. 



It has been proved, (p. 146), that the coefficient of the first 

 negative power of x in F(x) is equal to f e F(e e ).e e , taken from 6 to 



9 + 2x7- 1 or through one circulation. Applying this to the 

 theorems given in the former Sections, we get the .following 

 results. 



If <(> (x) = be any equation containing only positive and in- 

 teger powers of x, then 



2tt v /^Tx the least root * -J&e fl . 1. {#(rf).e -e J 

 2tt J - 1 x the sum of m least roots = -/ e e e 1. {<j>(e e ) .e-"' e }. 



