336 
EEV. E. HAELET ON THE METHOD OE STMMETEIC PEODHCTS. 
whence, multiplying, bearing in mind that f”=l, and arranging the multiplicand as 
below. 
we see that or 
=2'(^^2;^“}, 
which establishes one part of the theorem ; and the other part is established by simply 
interchanging m and jp. 
10. In order to illustrate some of the preceding properties of 2', let us take one of 
the factors 
/(a)=^j+a^2+a"^3, 
which enters into the symmetric product for cubics (art. 3). Then 'jrJ^x\ or 
y(“) • /(“') = 2' { {x, + ctx^ + cc^xT) } 
= 2a^^ + (a + a^)t’x^X2 
='Zcif—l,x^X2- 
Next, let us take the factor 
f{u)—X-^ + uX^-^-aJ^X^ -\-u^X^-\-ii)*X^, 
which enters into the resolvent product for quintics (art. 6). Then ‘ttJ^x)^ or 
2' { ^, (a^i + + ^^^3 + <»' ^4 + } 
= {2.2^+(iy -{-cu‘^)'^'x X2 -\-(co^-\-m^)2'x,X3} X 
{ 'Xo(f^-\-(^co^-\-ot)^y^’xyX2-\-{t*> x^x^} 
= ( 2.2^)^ — 2^2^iT 2 — ( + 5 2'.ri^22'^i^3 , 
which, by known relations among symmetric functions, may be readily put under the 
form exhibited at the foot of art. 6. 
I may notice here that if, in place of /(&;), we had dealt with/'"(ft;), that is. 
we should have been led to 
