312 OF STATISTICAL CONDITIONS. [CHAP. XIX. 



which are found in the development of the first member of the 

 above equation. Moreover, 0, and therefore 1 - 0, will consist 

 of a series of such constituents, having unity for their respective 

 coefficients. In determining the forms of the coefficients in the 

 development of the first member of (4), it will be convenient to 

 arrange them in the following manner : 



1st. The coefficients of constituents found in 1 - 0. 



2nd. The coefficient of x ly x z . . x tl) if found in 0. 



3rd. The coefficients of constituents found in 0, excluding the 

 constituent lc l , x 2 . . x n . 



The above is manifestly an exhaustive classification. 



First then ; the coefficient of any constituent found in 1 0, 

 will, in the development of the first member of (4), be of the form 



1 + positive terms derived from S. 



Hence, every such coefficient may be replaced by unity, Prop. i. 

 Chap. ix. 



Secondly ; the coefficient of x l . . z n9 if found in 0, in the 

 development of the first member of (4) will be 



SSrj 01 Si + S z + H n 



Thirdly; the coefficient of any other constituent, x l . . x i9 

 Zi+i . . x n , found in 0, in the development of the first member 

 of (4) will be *i . . + $j + $t+i . . + s n . 



Now it is seen, that E is the product of all the coefficients 

 above determined; but as the coefficients of those constituents 

 which are not found in reduce to unity, E may be regarded as 

 the product of the coefficients of those constituents which are found 

 in 0. From the mode in which those coefficients are formed, we 

 derive the following rule for the determination of -E, viz., in 

 each constituent found in 0, except the constituent xi x 2 . . x n , 

 for Xi write J 19 for Xi write s\ 9 and so on, and add the results; 

 but for the constituent Xi , x z . . x n , if it occur in <p , write ?i + J 2 . . + J n ; 

 the product of all these sums is E. 



To find E' we must in (3) make w = 0, and eliminate Xi , x, . . x n 

 from the reduced equation. That equation. will be 



-f X n ) S r +S r (Xr-Xt ..)} f = 0. (5) 



