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H 353 
THE SYMMETRIC FUNCTION TABLES 
OF THE FIFTEENTHIC. 
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My attention having been directed by Prof. W. H. Metzler, of the 
Syracuse University, to the desirability of having a complete table of the 
symmetric functions of the fifteenthic, I have completed the work here¬ 
with presented. Mistakes may have escaped the scrutiny of the computer, 
and should they be found he will deem it a favor to have them called to 
his attention. A brief historical summary of that part of the subject of 
symmetric functions which is connected with the computation and use of 
the tables is also given. Care has been taken to bestow credit where credit 
is due. 
HISTORICAL SKETCH. 
After the sixteenth century had produced solutions of the cubic and 
the quartic equations and had unsuccessfully attacked the solution of the 
quintic, attention gradually turned toward the relations between the roots 
and coefficients of equations. In fact, about the middle of that century, 
Yieta, an officer of the French Government, observed that in the equation 
x 2 + a 1 x + a 2 = 0 
if a x is the negative of the sum of two positive numbers oq, a 2 , of which a 2 
is the product, then a x and a 2 are roots of this equation. He failed to 
notice, however, the universality of the relation, as he recognized positive 
roots only. 
Knowledge of the relations between the roots and coefficients of equa¬ 
tions grew little by little until, at the time of Newton—when negative and 
imaginary roots had gained recognition—between the roots a 1 , a 2 , a 3 , ... . a n 
and the coefficients a x , a 2 , a 3 , . . . a n of the equation 
x n + a 1 x r '- 1 -\- a 2 x n ~ 2 H- +a r x n ~ r +- \-a n = 0 
the following relations were understood, namely 
2a 1 = a 1 + a 2 + a 3 +• • •+a n =—a x 
Sa x a 2 = a 1 a 2 + a 1 a 3 + • • • +a 2 a 3 + • • • = a 2 
S a x a 2 . . . a T = a x a 2 . . . a r + a x a 2 . . . a r _ 1 a r+1 + • • • = ( — l) r a r 
a x a 2 . . . a n =(-l) n a„ 
( 1 ) 
191800 
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