ccx Tables for Statisticians and Biometricians [XLIV 



second variate. While a whole series of equations like (i) (iii) can be written 

 down by varying n, and making p differ for X and F, they are of small service 

 without any further hypothesis, because n is in practice limited to a relatively 

 small number. A priori the most reasonable hypothesis appears to be that p- s , p +s , 

 p'-s, p'+s> P"-S> P"+S all rapidly decrease as s increases. An additional hypothesis, 

 which deserves in the first place consideration, is that the decreases in the three 

 series follow geometrical progressions, i.e. 



o / Ig II llg 



PS e PO> PS e ) PS ' 



Let us denote by < (n, e) the series 



( l) w ' '-' e' 1 + . . . H e 2 e + 1. 



Then we have 



= , 



= i-; ^_i = I 4 I ^ - (v), 



v & n+1 Y v } /. 2 \ 20(n + l, e)-l 



- =-^~ = 14 r ) ?rr7 ^ i 



V. n + I/ 2<p(n, e) 1 



(vi). 



Now if acj,, y v be the original variates including the secular trend, and we sup- 

 pose that trend to be parabolic in character, i.e. 



.+a s t s , 



then & n ac v = & n X v , A n y p = A n F p , if n be greater than s and s'*. Accordingly the 

 left-hand sides of (iv) to (vi) will be known if we difference the observed variates 

 and form the mean squares and mean product. 

 By interpolation into Table XLIV which gives 



4 ~ TT+T) ( 2<(n,e')-l 

 in heavy type, we can determine e, e' and e". 

 We then have 



w, e') - 1 V2</> (w, e") - 1 .. 



2 <f>(n, e)-l 



to find the correlation of X v and Y p , from the function 2^>(/i, e) 1 with the 

 appropriate values of e, e', e", as tabled in ordinary type in Table XLIV. 



Of course such values of e as are provided by the Table may represent only 

 roughly the real law of decadence, and their applicability should in every case be 

 interpreted with special regard to the data under investigation. 



* In practice it is easy to take the n different for the X and for the Y, if desirable, but a modifi- 

 cation of the function which represents {^. n X g A n Y s } in the present table is then required. 



