96 



Proceedings of the Royal Irish Academy. 



;ind tlie 2m -11^, and consequently depends on L, {%,n) and the T^. 

 It foUoTTS then easily that all integrals of the type 



r dz 



depend on Z (z, n) and the I^. 



All the elementary integrals, then, which can aiise in the dis- 

 cussion of the integrals 



4> (s) ch 



can be reduced to the 2m -11^ and integrals of the type Z[ , n 



so that ^\e need only consider these 2??^ forms when we discuss the 

 important question of the proper division of elementary integrals 

 upon which the general integral discussed in this paper depends. 

 0. If now, in the identity of (4) of Article 4, 



J/(^) 



-1^(3^^ dZ , .,, , ^ 



we let n become equal to a root aj, of j\z) = 0, it is clear the 



coefficient of -7— Tanishes, and we find 

 d/i 



(1) 



'^Jm 



'1 H (z, ai) dz 



-*/(«) ^1, 



f' 



where Z denotes Z 1 , ai ; from which we obtain 

 -1 i2(., aO J^i i/^ 



(2) 



Z 



c,/(-OJ/(^ 



, (. -aO/(aO 



It is evident that we shall have 2m of such equations, as there is 

 one corresponding to each of the 2tn roots of /(z) = ; and as these 

 2m equations enable us to express the 2m Z^, viz. Zj, Zj, . . . Zo„„ 

 in terms of the 2m- 1 I^, it follows that the 2m L^ are not indepen- 

 dent, but are connected by an equation which is easily obtained by 

 the elimination of the 2m- 1 Z^ in the following manner. 



6. It is not difficult to see that the highest power of aj in R{z, oi) 

 is 2m - 2, and consequently, if 2 denotes summation with regard to 



