Series to Frequency Distributions in Space. 



385 



Operating on each of equations 11, 12 with 6 and <f> and 

 putting x—y = l in the result, we obtain, since by (13), (14) 



[0(l-.fP + Q)] 1 = [-P + 0Q] 

 L*(l r rf , + Q]i = (*Q)i 



[tf(l-yB+S)] 1 =(«) 1 



[^(T^R + S)] 1 =[-R+^S] 1 , 

 the following equations : — 



— X20— %n— %iof^i — 3)+%oi(^ + l)— %ooO»2 — m t + r+2) 

 + ("-^')%2o + ^%ii + %io(— r + m. 2 + r' — n) = 0, . . . 



( n _ r')xii+^o2 + Xoi(~ *•— m 2 +r f -n)=0, .... 



-%o2-Xn-%oi(^i'-3)+Xio(r' + l)-%oo(w2'-Wi / + ^+2) 

 + 0' — »0Xo2+^Xn + Xoi(— ^'-W + r — /i) = .... 



z=y=l 



(15) 

 (16) 

 (17) 



(18) 



Solving these equations, we find after some reductions, 

 (g) i =^ I [(l + '-?)(l + '''?)»-7(«'' + '' + »-')-l], • (19) 



and these are the values of J -~ and *-fr. 



CC cr 



We can transfer these to the mean, since P2o=p'2Q—p'iQ 2 

 and p n =p'n—p'oip / iO' 

 The results are 



S) 1 %7^^1 + 3^ + r(r-l) ? *)-l-<,. + :>) ? ], . (20) 



_cV 9 (l- 9 )(n-r) 

 *»— („_1) -' • • 



_ «Vrfl-g)(»-Q 

 *■- („-l) -• • 



ccVo/1— 0) 



^■=--(,-17- ' • 



As shown in § 5 we may use Pearson's v 3 for 

 whence 



_ cV(l -g) (l -2g)(» -)•)(» -2)Q 



/>30 = 



>- 1)0.-2) 



/>03 = 



_ C 'V(l- 9 )(l-2 ? )(»-r')(»-2 



'•') 



(»- 1)0-2) 



• (21) 



• (22> 



• (23) 

 our Pm 



• (24) 



• (25) 



Phil. Mag. S. 6. Vol. 28. No. 165. Sept. 1914. 2 C 



