ON A NOVEL METHOD OF REGARDING ASSOCIATION 13 



and continuing the integration in this way by parts we have 



P= /2 1 (l-r^r""' 1 fi 11--^ 1.3 /l-r'Y 



N/irtr r 2(m + l)t 2(m + 2) r' "*" 2 (m + 2) 2 (m + 3) V r' ) 



1.3.5 



.m^-} 



2(m + 2)2(m + 3)2(m + 4) 



This series converges with considerable rapidity if* o- be not greater than "07 

 and r moderately large. If l/a^ = s, 2m = s — 3, and if we write X. = (l — r^)/r^, we 

 have 



P-_ p JZ 1 {l-^f'-'^ j. l.X , 1.3.x- 1.3.5.V 1 



Vtt"^ s-1 r Y s+l^{s + l){s + 3) (s + i)(s + 3)(s+5)^""J 



(xvi), 



whence P may be fairly easUy calculated. 



The series is a semi-converging one, which is satisfactory enough until r gets 

 small and therefore X large. When r is small or cr very large (xvi) fails to give the 



result closely enough*. In these cases the value of the integral I (1 —x^Y'dx must 

 be found from other considerations. Thus 



[ {\-x?Ydx= re'"'°s<^-^')(^x= ['e-^^'xe-™ (¥ + ?+•■•) c^a; 



J X J X J X 



= e (1— m — — m — + m^ — +...]««. 



Let ma;^ = ^z^ then 



(^ 1 (-J^ 12/1 1 1 \ 



(l-a;Tc?z = ^= e'* l-H^2'-J-.^° + 7lT^.z«+■••K«• 



Now the integrals 



-^Te-^'Uz and -^r e'^^'z^dz 



are tabled integrals, the first being the usual probability integral (Biometrika, 

 Vol. II. p. 182), and the second being the incomplete normal moment function which 

 has been calculated for n= 1 to 10 (Biometrika, Vol. vi. p. 66). Thus 



P = --/^, ,_g {h-o (n/2^) - N (V2^r)} - g^ V, (n/2^) - 1^, {J2^r)} 



- ^. {/*« (^/2^) - Ms (N/2^r-)} + j2g^, {/t, (V2^) - /., (y2^r)} - etc. . . .(xvii), 



1 f^ -I 2 

 where H'niz) = -7^ \ e *' z"c?z. 



«s/27r j 



* For most values of a- for r = -02 the two formulae coincide for practical purposes, so that the formula 

 to be given below may be used for values of r = -02 or under. For o- = -08, we cannot use (xvi) for r = 0-3. 



