Roberts — On the Reduction of an Integral, 97 



the 2??j forms described in the last article, we obtain, by the well- 

 known properties of partial fractions, 



'^V/W WfiS) v'/(2,)) 



as is otherwise obvious. 



If now we multiply equation (2) of the last article by a, and then 

 sum, we find 



(2) :§aZ = - 2(^-1)7,-21''-^, 



( W J (s) 



since the coefficient of a^"* is R{%^ a) is - 2(w - 1). 



It would then at first sight appear that by calculating 5a*Z, 

 2a^Z, . . . Sa^'^Z, we should obtain 2w - 1 equations sufficient to 

 determine the 1m - II, in term of 2»i - 1 independent I„ but such is 

 not the case as we shall now show. 



7. Prom the mode of the formation of the function Ii.(z, w) it is 

 easily seen that it cannot contain any term of the form z''n'', but the 

 only term whose disappearance is of consequence is s"' ~ ^w"' ~ ^, for it 

 follows from the non-appearance of this term that we are unable to 

 determine /„_i ; and it becomes then possible to obtain another equation 

 connecting the 2wi Z^, so that, in general, we must regard but 2?;i - 2 

 of the Zj as independent. 



It is impossible consequently to express /„_i in terms of the I„ 

 but we can express the remaining 2m - 2 /, in term of the Z^ and the 

 integral Z„_i. 



An example will make this more clear. Let us take tn - S, and 

 write down R (s, n) which is easily found to be 



R (s, n) = 4z* + 8P,z^ + iP^z-" + F^z + n {2z^ + PjZ^ - J>^) 

 - 71" {P^z + 2P2) - ?i' (22 + 3Pi) - 4n*. 



Introducing the value of R (2, n) into the formula 

 Z = 



^(^.4__o/' v//(2)__, 



(1) 2Z = - 2 



we find easily 



1 



(2) 2aZ 3. - 4Zo - 2 



^,v//(3) 



E.T.A. PROC, SEE. UI., VOL. TI. 



