198 REV. R. HARLEY ON SYMMETRIC PRODUCTS, AND 
~ [a] [6 +8] [y] + 2[a] [8-+7 +8] - [a+ 4 fv [8] 
~ [a+ y] [6118] - [a+ 8][A][o] + [a+ A) fy +8] 
+2[a+7+8][6]+2[at+h+y] [6] + [a+9][B+y] 
+2[a+B+8][y]+ [a+y]}[8+8]-6la+8+y+6], 
(aBor”) = 44 [a] (8) Ly]? [2] [4] [27] - 2[a] 8+] fy] 
+2[a][8+2y] - [+A] [y]’-2[¢+ 7] [4] [7] 
+ [a+] [2] +2[a+ 27] [6] +4[a+ 8+] [7 
+2[a+y][B+y] -6[¢+8+2y]§, 
(aap) = 34 [a}*[A} - [a]*(28} - 4] [a +8] [6] 
+4[a][a+26] - [2a] [8]*+ [2a] [26] 
yelled [8B] +2[a+]*-6[2a+2P}}, 
(a888) =¥ [a] [8]* - 3[a] [8] [28] +2[a] (38) 
~8[a+A][6]'+8[a +8] [28] + 6[a +28) [6] 
~ ee t, and 
(aaaa) =3;} [a]*— 6[a]”[2a] + 8[a] [8a] + 3[ 2a}? 
— 6[4a]}. 
Hirsch denotes repeating exponents thus: 
Sze at by (a*), Sarat xh xf xf by (a*6"), &e 
but I have thought it conducive to clearness to express 
them as above. It will be observed that the crotchet [ | 
is employed to represent the sum of powers, and the 
parenthesis () that of other functions of the roots. 
By the aid of the formule in this article and the values 
of [1], [2], &c.... [20], given in the last, we may easily 
calculate the value of any symmetric function of x in 
terms of Q and E. 
35. Thus taking the functions that occur in the value 
of oH va 31), we have * 
4) = 44 [4]*-[8]}=430°-(-8-5QE){=4-5QE, 
=[s01)= [413101 - (7101 - ‘ [3] - [4° +2[8] 
—5QE, 
5 (422) =41[4] (2]°-2 [6] [2] - [412+ 218)}=-8-5@, 
¥(8311) =44 (3])°[1]?- [3)°[2] -4[8] (4) (1) +4(3] 5] 
* To prevent confusion, the symbol 3 is prefixed to the parenthesis (). 
