282 REPOiiT— 1875. 



I proceed to explain the Subsidiary Table, first in its application to 

 Table I. 



The Subsidiary Table is divided into sections, giving the GF's of the suc- 

 cessive columns of Table I., each section being given by means of the pre- 

 ceding columns of Table I. ; for instance, that for column 3 by means of 

 columns 0, 1, 2 of Table I. 



As regards column 0, the Table shows that the GF is =x. 



As regards column 1, that the GF has a first factor, 



(l-tx)-\ ={l) + tv + fx' + t\v'+ ..., 



which is operated on by the symbol [<'•••'"], viz. the constant term (1) is to 

 be rejected ; and that it has a second factor, = ,v : the product of these, viz. 

 {tv+t-x:--\-fx^ . . .) X X, is the required GF, the coefficients of which are ac- 

 cordingly given in column 1 of Table I. 



As regards column 2, it shows that the GF has a first factor, 



(l-te=)-i (1-tv')-^ (1-te*)-' . . . ., 



where the indices —1,-1, — 1 , . . are the sums of the numbers in column 1, 

 Table I., which first factor is 



1 + tv^+tx^ . +r t \x^+ . . . 



(and it is as before to be operated on with [<^ " " ' °°], viz. the constant term is to 

 be rejected) ; and further, that there is a second factor = x+tv- + t-x^+ . . ., 

 the coefficients of which are obtained by summation of the numbers in the 

 several lines of columns 0, 1 of Table I. We have thence column 2 of 

 Table I. 



As regards column 3, it shows that the GF has a first factor, 



(l-te')-i (l-tx*)-'' (1-t.v')-* ...., 



where the indices — 1, —2, — 4 . . are the sums of the numbers in column 2 

 of Table I., which first factor is 



=l+tx^+2tx^+U 



'3:'+ ( Qt\cc^+ ... 



(and it is as before to be operated on with [i^""°°], viz. the constant term is 

 to be rejected) ; and that there is a second factor 



te^+y t \x^+ / t \ x*-\- . . ., 



the coefficients of which are obtained by summation of the numbers in the 

 several lines of columns 0, 1, 2 of Table I. : we have thence column 3 of 

 Table I. 



And similarly, by means of columns 0, 1, 2, 3 of Table I., we form the 

 GF of column 4 ; that is, we obtain column 4 of Table I., and so on in- 

 definitely. 



To apply the Subsidiary Table to the calculation of the GF!s of Table II., 



