1905 - 6 .] Dr Muir on the Theory of Alternants. 
383 
s o 
s x • . 
• • S n _i 
1 
So 
Si • 
• • • S n _i 
Si 
s 2 • • 
. . Sn 
X 
= (x n — ppc n 1 + . . 
• •) 
Si 
s 2 • 
• • . «« 
s n — 1 
«» • ■ < 
. . S‘jn—2 
x n ~ l 
s« 
S w +i • • ■ 
■ • s 2n- 1 
x n 
Sn— 1 
Sn • 
. • • S 2n — 2 
a result already reached by Joachimsthal, and which by the 
equatement of like powers of x gives “ i coefficient p espressi da 
rapporti di determinant di n esimo grado.” 
Betti (1857, June). 
[Sur les fonctions symetriques des racines des equations. 
Crelle’s Journ , liv. pp. 98-100.] 
Betti recalls Borchardt’s result of the year 1855, namely, that 
the symmetric function 
2 
1 
X 1 z 
n 
where x l , x 2 , . . . , x n are the roots of the equation 
0 = x n — pps 71 ' 1 + p 2 x n ~ 2 — . , 
=f(x) say, 
is the coefficient of 2+1 > . . . t~ in the develop- 
ment of 
/ JL ± a r n(<! , t 2 , . . . , t„) \ 
y ' lift 3 3 u a Wi) • /(<„) . . . /(<„)/ 
according to descending powers of the r‘s. He then gives an 
observation of his own, namely, that the said symmetric function 
is likewise the coefficient of tf a i +1 )/ _ ( a 2 +i) . . . £-(®»+i) j n 
similar development of 
/(*. )•/(*»)•••/(<■.)• n»(< . 
‘ fifi) • • • fi^n) ’ H 2 (aq, x 2 , . . . , x n ) 
and by comparison of the two results draws the conclusion that if 
Borchardt’s generating function be denoted by 
6(ti , t^ , • • . , ^n)j 
and his own after removal of n 2 (aq , , . . . , x n ) from the 
denominator be denoted by 
, t 2 , . . . , ^ n ) 
