») 



Function of any Number of Variates. 447 



found to be very useful in discussing the mathematical 

 properties of the function. Indeed we have 



<£(>!, tf 2 , .... Xn) =Tj-\ n \ dw l \ dw 1 • • • • 



^d^e-— "■"•'■'"'"" e-'-^". (2) 



%y -co 



This integral, which for w=l reduces to the well-known 

 integral of Laplace, has been given by Charlier for the 

 special case o£ n = 2. A proof of the theorem for any value 

 of n will be given in the Appendix. 



By partial integration it is further easily verified that 



g* t +* 9 +. • • -+^.r^4 2 — •^■^{«h ** — *•)= 



= — — \ ^1 \ dw * 1 ^' tt -^ h^ K 7^ * 



(27t)^J-» J-« ■>-- B^PBw? Bm£" 



Hence by the theorem of Fourier we have 



J -co J-» J— « 



Putting 



; ^^p^f^ = H K> „ 2! .... (tv) ,-n**^ 



we find, as 



H (o, o, .... o) = o 



A^AT., .... A'n, 



when k x + k 2 + + fa is an odd number, for the 



moments of any order of <j>(z u oc 2 , x n ) the formula 



d#! 1 <fa S 1 rfa? » #lW &»" </>Ol, #2> #n) 



— x * J -x J — «° 



= (-1) 2 "H (0,0, ....0). (4) 



k-Jc^ .... /u 



