A MATHEMATICAL THEORY OF RANDOM MIGRATION 11 



Then it is clear that Xn> if 9. he less than s, consists of powers of r 2 less than 

 s, and therefore 



/: 



a) 2s x m rdr = 0. 



Accordingly a remarkable property holds for the ^-function part of the co-function, 

 namely, if Xzq an d X25' De two su ch functions, then it follows that 



/; 



e X^Xm' rc ^ r ~^> ^ 1 an d 9.' De different, (xix). 



Returning to (xvi), let us put q = s, then 



m 2Si2s = -(2a>y\s I ™p- s - 2 e™dl3 

 J -0 



J^^- i-iyfx^e'^dx, 



or, m^ = (-iycf«2*(\sy (xx). 



Let us now consider the integral over the plane 



1= 2tt I v^XisTdr. 



Except for the last term in x™> it will consist of a number of terms having for 

 factors m M2g with q<s and these all vanish. The last term in ^w is 



and accordingly 



I=2tt\ ^ x , s rdr= 2lT \ s > -g co 2s r 2s+1 dr, 

 Jo * & j 



or by (xx) 7=(|s) 2 (xxi). 



Hence we have the following properties : 



(a) The integral all over the plane of distribution of one product of a 

 ^-function into an co-function of a different order is zero. 



(b) The integral all over the plane of distribution of the product of a 

 ^-function into an co-function of the same order is, if 2s be the order, equal 



to (\sy. 



These properties enable us — as in the case of Bessel's functions or Legendre's 

 functions — to expand any function symmetrical round a centre and a function 

 only of the square of the distance from that centre in co-functions. 



Thus let F(r*) = S ls (b 2s o>, s ), 



s=0 



2—2 



