ON MATHEMATICAL TABLES. 



323 



Table II. (6 pages). The successive lines give the coefficients in the ex- 

 pansions of 



~ S S S 



•where 



1-0,-' 1- 



S=- 





-.v.l-.v\...l-x~" 

 ad. iuf.. 



each expanded as far as af^, and further continued as regards the first ten 

 lines, that is, the expansions of 



™ s s s 



-af 



l-.v.l-w'' 1- 



-x" 



1-x' l-.r.l 



each as far as .r'"^. 



Table III. (2 pages). The successive lines give the coefficients in the ex- 

 pansions of 



S' S' S 



S^, 



1-cr.l- 



0.5' 



each expanded as far as a'". 



36. As regards Tables II. and III., the analytical explanations have been 

 given in the first instance ; but it is easy to see that the tables give numbers 

 of partitions. Thus in table II. the second line gives the coefficients in the 

 development of -^ 



(l-x)Xl-x'Xl-x').... ' 



viz. these are 1, 2, 4, 7, 12, 19, 30 .... , being the number of ways in which 

 the numbers 0, 1, 2, 3, 4, &c. respectively can be made up with the parts 

 1, 1', 2, 3, 4, &c. ; thus 



Partitions. No.= 

 2 



1 

 1' 



2 



1,1 



1,1' 



r, 1' 



3 



2,1 

 2,1' 

 1,1,1 



1, 1, r 

 1, r, r 

 r, V, 1 



&c. 



&c. 



and similarly the third line shows the number of ways in which these 

 numbers respectively can be made up with the parts 1, 1', 2, 2', 3, 4, 5, &c ; 

 the fourth line with the parts 1, 1', 2, 2', 3, 3', 4, 5, &c.; and so on. 



And in like manner in Table III. the first line shows the number of ways 

 when the parts are 1, 1', 2, 2', 3, 3' .... ; the second line when they are 



1, V, 1", 2, 2', 3, 3' ; the third when they are 1, 1', 1", 2, 2' 2", 3, 3', 



&c. ; and so on. 



x2 



