232 PROFESSOR K. PEARSON AND MR. L. N. G. FTLON 



Taking logarithms, the products II become sums S, or 



log (P A /P ) = S log/fa, + A/t,, x 2 + A/i 2 , . . . x m + AA m ; c, + Ac,, c 2 + Ac 2 , . . . c p + Ac,, 1 ) 

 S log/(,, a;,, . . . x m ; c,, c 2 , . . . c ;) ). 



Let the first summation now be expanded by TAYLOR'S theorem, and typical terms 

 up to the second order be written down. Then we have 



log (P A /P ) = AA, S (log/) + i (A>0 2 S ~ (log/) + AA, A/ iy S ^ (log/) . 

 + Ac, S ~ (log/) + 1 (Ac s )* S ~ (log/) + Ac, Ac,- S ~- (log/) 

 + A/t, Ac, S -n-y- (log/) + . . . + cubic terms in A/i and Ac + & c -> 



where /stands for/(x,, o; 2 , a; 3 , . . . X M , c t , c 2) . . . c^,). 



Here r is to be given every value from I to m, and * to be given every value from 

 1 to p, but r' and s' in the third and sixth sums are only to be given values from 1 

 to m and l.to p other than r and s respectively. In the above formula we may 

 replace the sums by integrals, if we remember that the frequency of the system 

 x,, x 2 , . . . X M , is simply /So;, S.r 2 . . . So:,,. 



Writing 



log (P A /P ) = A, Mi,. - IB, (A/i,.) 2 + C,,, A/i, Mi, + D, Ac s - P, (Ac 8 ) 2 + F M , Ac, Ac, 



we will investigate the values of the constants separately. 

 First, 



= JJ| . . . ^- dx l dx 2 . . . dx m , 



= III [/] ^1 ^2 ' <&r-l rfaj r + l dx m , 



the integrals now not including one with regard to dx r and [/] denoting that / is to 

 be taken between the extreme limits of / for x r . Now in most cases of frequency 

 the frequencies for extreme values of any organ are zero.* Hence [/] equals 

 nothing. Thus we have A,. = 0. 



* In most cases, but not invariably, as, for example, in the case of some florets and petals. In the 

 cases, however, in which A r does not vanish, the conclusions finally reached will be the same, as A r only 

 marks a change of origin for the constant frequency distribution. 



