and 



( 347 ) 



Pl,2,„-l (u) = — (6iM + hy + . . . + b2m-\ «2'»-l) 



are a pair of Schlafli's polynomials. 

 Using this relation we obtain 



00 



and as 



6„ = Lim «2»» Py 2,„ (y/) = 22'« ??i ! 



we have 



22'" m ! = o2/«+i L^o„j j^^^^_|_j (,,^.) j^ (^<«) j^ („^j . . .Ja ("rt«-i) c/w 







with the conditions 



c>rt + «1 4- . . . +a„_i , m<— — . 



Evidently the value of the integral would be zero, if instead of 

 the first of these conditions the condition 



« > c + «1 + • • • + ««-I 

 was satisfied. 



In the same manner we might differentiate and also integrate with 

 respect to one or to several of the parameters a. This leads for 

 instance to the following results 



n even : z= i J^ (jic) J^ (na) J^ (w«i) . . . J^ (?/a„_i) chi 







CO 



n odd : zrz t u J^ (uc) J^ {ua) J^ {ua^ . . . J^ (wa„_]) du . 







c]>a-f «J 4-^2 + ••••+ «»«—!• 

 Still other results present themselves when Pearson's problem is 

 slightly modified. Again putting 



J„ {ua) J„ (Mtti) J„ {uan-\) =f{u) 



and writing 9 for c, we get by differentiation with respect to q 



