G. U. YULK 123 



Hcuce it is quitc unnecexsart/ to statu ((// the f're(iuencies as iiih1l-i- (I); if 

 Space is of iniportance iio more need be giveii than the cight frcqueiicies of the 

 third Order, the uUimate frcqueiicies as they may be termed (i.e. frcqueiicies 

 of classes specified by the whole iiumbcr of attributes noted). The numbcr of 

 frequencies in an aggregatc of order n, i.s cvidently 2", so that not nioru than 

 2" frei]iU'nfies need bo stated in any case whci-e n attributes are observed. 



Any othor set of 2" inch'pendcnt fro(|uciicies may however be chosen instead 

 of the 2" ultiniate frecjuencies, the set fornied by taking N together with the 

 frequencies of all tiie positive classes offering several advantages. It is not 

 difficult to See that any frcquency whatever can be expressed in terms of the 

 numbor of observations and the positive-class frequencies by using the relations 

 (2). We liave for instance 



(a) =N^(A) 

 (aß) ={B)-(AB) 



{aß) =ia)-(aB) = N-(A)-(B) + (AB) ^ (3), 



(a^7) = (aß) - (aßC) = (aß) - (aC) + {-xBO) 



= N-{A)-{B)-{C) + {AB) + {AG) + {BG) - (ABC) 

 and so on. 



To take a very simple e.xamplu with two attributes only, consider the State- 

 ment of results of one of Mr Bateson's e.xperiments on hybridisation with poultry, 

 Leghoru-Dorking hybrids when crossed inter se produced offspring of varied fornis ; 

 some having the rose comb and some not, some having the extra too that 

 characterises the Dorking and some not. Using A to denote "rose comb, " B to 

 denote extra toe, the numerical results may be completely expressed in either of 

 the following fornis 



I II 



(AB) = 208 N = 336 



(Aß)= 63 (A) =271 



(aB) = 54 (B) = 262 



(aß) = 11 (^i?) = 208. 



The advantages of the second form of tabulation are obvious ; it gives at sight 

 the whole number of observations and the numbers of ^'s and B'a. The first 

 table gives neither without reckoning, yet both are equally complete. 



A rather interesting case arises where the frequencies of contrary classes are 

 equal, as may be the case if the character dealt with is really variable and the 

 points of division between ^'s and a's are taken at the medians. Such a condition 

 implies necessary relations between the class-frequencies of any odd order and the 

 frequencies of next lower order, l)ut for the discussiun of the case I must refer to 

 the first niemoir mentioned in iiiy note § on p. 121. 



16—2 



