14 KARL PEARSON 



whence integrating by parts : 



3 



^vM-^H)^ 



sb( s -£>MA ; -j 



i l( s _I)( s _')( s _ 5 



V2iro-2\ 2/\ 2/\ 2 



2/\ •: 



4f "/8-V"dj8 



2J -o 



2Jo 



• (xxv), 



or > 1. 



If s = 0, 

 I now take 



, 1= 1 * 

 ° v/2tto- 2 



.(xxvi). 



^2p, 2« " 



(o,y p dr 



-wj. •*<-»" j$(H*- 



and find, reducing in the same manner, 



P.2S " 



: 7S¥( s - p 4)( s - ? -2)-(- J,+ i) ><1 - 3 ' 5 - <2f, " 1) '" (xxvii) ' 



Clearly : 

 by (xiii), hence : 

 Or, 



m !j)-l,!S- 2 



7r cv^dr 



m 2p _ 1 ^ = -j27rcr' p " 1 [s-p-^J (* ~P ~ jj 



-p + - 



.(xxviii). 



x 1.3.5 (2p — 1) 



Thus the odd moments of the w, s functions are known*. 

 For the particular case when p = 1 : 



^ = 72^?H)H)---(- 2 : ) 



if & 2S be the swing-radius round the axis of the function co. 2s . Hence by (xxv) 



U=-hf=-2Tri ^ XXU1 )- 



.(xxxii), 



* If x = r/cr the following finite difference and differential equations are fundamental in the 

 theory of the co-functions : 



<»2< s +2) - (2* + 3 - iz 2 ) o) 2(s+1) + (s + 1 ) 2 w 2s = (xxix), 



do: 



1) 2( S +i) = (* + l)^ s + i a; -^ 



(xxx), 



dx? 



(1 \ dm.,, 

 X+ x)^x +2is+l)w ' s = ( xxxi )- 



But the fuller treatment of the to-functions must be' deferred. 



