143 



where C) are the binomial coeflicieiits. This relation may be obtained 

 in the following way. 



According to Art. 8 II we have 



,_o,,_,. ^ 1 _ /, n^ (,) + ^ n, {z) - I ƒ/, (.) + {p) 



and, expanding by Taylor's theorem 



F {x + k, y 4- k) 



where 



F (.t-, y) = c—^"—f 



g-(.r+tr— r^/+^•)^g^-+.'/"=:f3:2+v2 



p—X — .?/ 





J — 



' 2/ 



''■''£ G-"0+'<s ('"0 è G~0+^"'' ^ ' ^"" '' +• 



(i.i-^ 



which may be written 



p—2k x—-2k //— •2^•2— 1 . 



k[n,{,r) + n,{y)]+-yi,{x)+2H,{.v)n,{y)-\-H,{y)]+...{q) 



X -\-y 7 '^ 



Putting now z =: —=- in {p) and X; = -7= m (^) w^e get 



.r Vy\ h' /.r-f-t/ 



^ Vl/2 



9/ 



1/2^ 



,-:..+,)u -i-/-^ =,!._— [//^ (.,) + n, 0/)] + 



+ —=- Uh (-^0 4- 2 ƒ/, (.r) //, (y) + ƒ/, (t/)] + . . . 

 (|/2)''2. 



A" 



Comparing the coefficients of — m tlie second members we obtain 



n! 



tlie required relation (31). 



Proceeding to the reduction of the integral 



M=z 



je--^'iLU^) 



x)e-i"+-^r- n„{i(^x)dx 



we put, according to (2) 



d"> 



cfx'" 



then 



Now, integrating by parts we have generally 



