Ah. A. TCHOUPROFF 
169 
When t the coefficient of in the development of Ef.,j+i) dfju consecjuently 
reduces to zero. When ^ = i + 1 , noting that 
(- 1 C!:,_^ = i J^ic,, = 1.3.5... {-Ik - 1) !(/ _ Z: + 1) + ^ (/• - 1 )} 
// ~ 0 /7 = 0 
*Zk 1 
= 2 (-iy'6'::,_,^,-+,_„,,, 
/(=0 
we find : 
(-l)'«].3.5...(2^+l)f^^l,2/+u ^ 
+ (- 1/ 
*=1 
''a.3.5...(2/,:-]){ 
n 
(21+1) 
1 - '-^(-2/ 
il/' 
..2. 3 
if 
(/-7f + l) 
, ...2, :! 
whei-e mI\J'^]^ ^ has the vahie o-iven above, and il'/ ' denotes an analogous 
sum of terms of type /j,/,^ yu,/,, ... u/ii^j- witli the single difference that in the aggre- 
gate there occur not two but three of the numbers 77^, iv^, ... '/2j_2i.-+ii while 
the remaining 2i — 2/,' — 2 numbers of this series are distributed in pairs among 
/'l, /'2, ••• 
In the case = /-o = . . . = jv+i , the coefficient of l/i\"''+' in the development of 
-^(2i+i) c^/" becomes : 
(- 1 )'•+• 1 . 3 . 5 . . . (2^• + 1) il / + (- 1 1 . 3 . 5 . . . (2i - 1) [1 + 2 ( i - ^ )] f^l+i /"/"V^r 
+ [ii-I,+ }) + iik-\)] r?';;f ^V''-' ' 1 . 3 . 5 . . . (2. - 2/- + 1 )| 
+ 0;!, + ,1.3.5...(2/-3) 
= 1 . 3 . .') . . . (2i + 1) .-, [m,^ - /A,.-J'"' . [fM,r - 3//,, /I, + 2^./']. 
(Cf (If)) above.) 
(To be continued.) 
