30 DR GUSTAVE PLARR ON THE 



Let us apply Gauss's method to this particular case only. We have then 



( a +iy/+i = a '/+i°^. 



Hence 



(a + iyi+ l -a t f +1 t g'/+ 1 

 1*1+1 a l'/+i- 



This relation takes place for any value of t. For i = as well as for 0>t x (where 

 t x = — a) both members vanish. We may put the second member under the form 



and make use of it only from t = 1 to t = t x ; or, if we put 



t-l=t', 



make t' to vary from t' = to t' = t l ' = t l —\ (where a+ 1+^— 1 = (a+ 1 +*/) vanishes). 

 Thus we may write 



(g+l) t/+1 -a t/+1 _ (a + l) t 7+1 



Likewise we have 



/3'/+ x =/ 8 (/3 + 1//+ 1 



7 V+1_ 7 ' (7+iy 7+1 ' 



Multiplying this member to member with the preceding equality, and summing in the 

 first member from t = to t = t u in the second member from t' = to t' = - a - 1, we get 



( a ) f( c L±h£)-f(?iM)=P ffSi+ldttl) 

 Secondly, in treating ft as we have treated a, we get 



/(^)-/(fH/( aJ ^)' 



We write this result for a+ 1 throughout, in the place of a. This gives 

 W // a + 1 -'./3± lN ) f/a+l,p\_a + l f(<* + 2Jl±l\ 



In the third place, we have 



1 _ y-l+t 1 1 P+l 1 



( 7 _iy/+i 7 _i "yH-i-yH-i ' y-i^y+iyv+i- 

 Hence 



j_r_JL__j_i = _i i . 



r /+1 L(7-i)' /+1 y /+1 J (7-i)7-' r /+1 .(7+i)' /+1 



We have also 



