368 Mr. C. E. Van Orstrand on 



The polynomial theorem gives 



(b 1 x+b^+...Y^j^pi--^ v+H+ -, ■ (6) 

 the exponents being subject to the conditions 



p + q +...^r, | . .... (7) 



p + 2g+...=n-l.) 



The first condition arises from the homogeneity of the b 

 terms in the expanded equation, and the second is imposed 

 by the condition that the terms in sc must be of degree 

 {n — 1) in order that the complete expansion of y~ n contain a 

 term in a~\ These conditions therefore require that the 6 

 terms are of order r and weight (n — 1) instead of order 

 (r + 1) and weight (n — 2) as in expression (a). This dif- 

 ference in the order and weight is due to the fact that (a) is 

 the coefficient of z n ~ l instead of z n . Finally, by substituting 

 (6) in (5) and (5) in (4), we obtain 



K __ 2 l n + lXn + 2)...(n + r-l ) ^ t (g) 



as the coefficient of ?/ % a~ n = ^ 1 . 



Formulas (7) and (8) hold for all positive integral values 

 of r and n. The seemingly exceptional case, r=l, is readily 

 seen from (4) and (5) to give a numerical coefficient unity 

 for all values of n. Since the b terms are of a given order 

 and weight, they may be taken in part from tables, such as 

 Bruno's table, " Symmetrische Funktionen der Wurzeln 

 einer Gleichung," contained in his treatise, 'Binare Formen,' 

 which contains all terms of successive orders and weights 

 from 1 to 11 inclusive. Terms of weight 12 may be deter- 

 mined with the assistance of the same table, for evidently 

 the expression, b x x terms of weight ll + b 2 x terms of weight 

 12 + ..., contains all terms of weight 12, including dupli- 

 cations. The process may be continued so as to determine 

 all terms of any given order and weight *. 



Having determined p, q ... in the manner indicated above 

 for any particular value of n, the corresponding coefficient 

 is readily determined by substitution in either of the preceding 



* For a precise method of determining order and weight, see a paper 

 by R. A. Harris, " On the Expansion of Sn #," Annals of Mathematics, 



1888, p. 87. 



