G. U. YULE 129 



Bat all tlio f'reiiuencies ou tho right are giveii hy tlie rclations of iadependence. 

 Thereforc 



{AßyB...^u)=^/^^ (iV- -(/i)(C)(Z))...(Ji)(i\0 



-(B){C)(D)...{M){v} 



-iB)(C)(D)...(f.)(v) 



-(B)iy)iS)...(f^){v) 



- )• 



Now consicler the terms on the right in tho bracket. With the exception of 

 the one tena (ß) (y) {S)...{fj,){i>), the remainder can be grouped in pairs of which 

 the one moiaber contains (B) and the other (ß), the following frequencies in each 

 niember of the pair being the sarne. Carrying out this rearrangement the 

 expression will read 



iAßyS...i,v)=jß-^ jiV«-'-(i?)(.y)(8)...(M)W 



-iB){C){D)...{M){N) 

 -{ß){C)(D)...{M){N) 

 -{B){C){B)...(M){v) 



-{ß)(C){D}...{M)(v) 



-(B){C)i8)...if.)(v) 

 -(ß)iC){8)...(f.)(v) 



- )• 



Replace (B) by N - (ß) throaghoiit and rearrange the terms in similar pairs 

 containing C and 7. (B) and (ß) are then eliminated from all the terms and 

 the expression then becomes 



{AßyS...f.v) = ^_- {iV- + (ß) (y) (8) . . . (f.) (v) 



-N(y){S)...(^)iv) 

 -N(G)(S)...(f.){v) _ ,_ 



-N(C)(D)...(M)(N) 

 -N{y)iD)...(3r,(N) 



-N{C){D)...{M){v) 

 -Niy)iD)...{M)(v) 

 - !• 



Biometiika 11 17 



