10 KAKL PEAESON 



etc (xii). 



The w-functions form a series of such special interest that a few of their 

 remarkable properties will be stated in the next section. 



(4) Properties of the a- functions. 



Let us consider the j>th moment round the origin of the 2.sth w-function taken 

 over all plane space. We will denote it by m PiSg . Then 



f 2ir f °° 

 m p,u= dd rdroi M r p 



— 2tt\ h>. 2s r p+1 dr (xiii). 



Jo 



Now -—d*i-er'$$"') <")■ 



and by fi= — 2cr 2 /r 2 we have clfi = — dr. 



Hence writing p = 2q we find 



«^.=(-ir'-H*oy f^p-'-^^rfdp (xv). 



Integrate by parts and we have 



», 1 ,„=(-ir.-.(2^[{^-.|^ l (ie.»)};- (s - 9 -i,|;>-.-.|^( i i^^ . 



The part in curled brackets vanishes at the limits and thus 



=^ ?>M - 2 (s-g-i). 



Repeating this process we find 



m 2gi2s = (.s-l- 2 )(s-2-g)(s-3- 9 )...(- 2 ) 



x(-l)«- 1 x(2o- 2 ) 3 x f "p-^e^dp ( X vi). 



The integral is finite and known ; hence if q be less than s we find for integer 

 values 



'«**,.* = 0, g<* (xvii). 



Now consider co M as made up of two parts, 



^W* " X *» (xviii). 



