328 
BEV. E. HAELEY ON THE METHOD OF STMMETEIC PEODTJCTS. 
Let Xj, Xa, X 3 , . . . X„_i be linear unsymmetric functions of x and of the form 
where the n—1 constants ^ 2 ? •• • arbitrary. Then if n be less than 5, the 
constants which occur in the 1 functions may be so distributed and determined as 
to render the product 
1(^) XjXaXa . . . !X„_j 
(or, ■when n=-2, X^) symmetric relatively to X’, but if n be equal to, or greater than 5, 
the symmetry is not in general attainable. The product is called the symmetric 
or resolvent 'product according as it is or is not symmetric. When this symmetry exists, 
7r„_i(x) can of course be expressed as a rational function of the coefficients a, b, c, &c. 
When it does not exist, we form the analogues of the functions which enter into the 
symmetric cases, and the symmetric product Il(x) is then obtained by multiplying 
together the several unequal values which the resolvent product ‘r,,_^(x) can be made to 
take by the permutation of the ar’s ; and n(a’) may, in this case also, be expressed as a 
rational function of the coefficients. 
2. The simplest case is the quadratic 
(a, b, c'Xx, lf=a{x—Xi){x—X2) = 0. 
Here 
{‘r,{x)y=X.^=(x\+kx2)\ 
and it is seen at a glance that when k= — 1, X is unsymmetric and X^ is symmetric. 
Hence 
X^= (%xf — 4 X 1 X 2 =~(P— 4ac), 
and the roots of the quadratic are 
—b±a7ri(x)} =^(—b b^ - 4ac). 
3. The next case is the cubic 
(«, b, c, d\x, lY=a{x—x^){x—X2){x—X3)=^. 
-X] — Xi ~j“ TC1X2 “h 5 
-Xj 2 — Xi I ^2^2 k^x ^ , 
^ 2 (^}~XjX 2 , 
Assume 
and 
tion (a, h, c, ... ^)” would be written 
+ &c., 
and (a, I, c, ... y)” would be written 
ca;"~^y^+&c. 
