182 REV. R. HARLEY ON SYMMETRIC PRODUCTS, AND 
(a® — 4ab + 8c) p> — AG c+ 8ad — 4bc)p* 
+ 5 (4abd — ac* — 8cd)p + i (8ad* — 4bed + c*) =0. 
Hence p may be determined; A, B, C, D and conse- 
quently y become known, as before; and z is given by 
a= — ft apy - Up + bpp 
y* + apy? + (bp —2)py + ep? — ap?’ 
a result obtained, like the corresponding one in the last 
article, by the aid of the common divisor. 
Secrion II. 
The Resolvent Product for Quintics, and a new Cyclical 
Symbol. 
15. For the general equation of the fifth degree, 
xv + ax + ba + cx*+dx+e=0, 
the resolvent product is (Art. 5) 
1 4(@) = (Dax)* — 5(Lax)? Day v.45 (La, 2)? +577 
=a'—5@6+56?+ 517’. 
This product is six-valued. For 
SO=C a t+ia,t+Pat+ Patia;, 
and therefore f(i) has 1-2-3-4-5 or 120 values, which 
may be formed either by permuting the powers of é or the 
roots of the quintic in all possible ways. Let us call f(z) 
the first factor of 7,(#). Then each of the values of f(i) 
may be made a first factor, so that, at first sight, it might 
seem that 7,(#) has 120 values. But by the properties 
of 3, 
m2) =f) SE) L® fe) 
Whence it appears that when /(i), f(#), or f(#*), becomes 
the first factor, no new values of 7,(x) are introduced, but 
we are remitted to our former value. Thus the number is 
reduced one-fourth. Again: — Let 
