THEIR APPLICATION TO THE SOLUTION OF EQUATIONS. 199 
~ [6]{1]° + [6] [2] +4(7] 1) +2[4]°-6(8]3 
=-—3-5QE, 
S (8221) =4§ [3] (11[21*- (3) [1] [4] -2(8]°[2] +2(8) [5] 
~ [4] [2]°-215] (1) (2) + [4° +2710) +46) (2) 
42[5] [3] -6[8]}=-6-5QE, 
5 (2222) =z f [2]*-6[2]°[4] +8[2] [6] + 3[4]*-6[8]} 
—2.5Q5, 
¥ (22211) =-ES(111)=-F {1 }*-3[2] 21) +2(3]} 
=—5QE; 
- Sr#=8¥ (44) + 43431) + 63 (422)) + 83 (3311) 
443 (3221) +123 (2222) + 365(22211) 
=— 4.5°QE. 
Also, 
(41111) =- E[3]=-3-5QH, and 
+, SU%= 23 (422) + 125 (41111) +45(3311) + 45(3221) 
4123 (22211) =— 4.5°QE, 
which coincides, as it ought to do, with the value of 3'r*. 
36. Now (Art. 32), 
D,=-}30=- 3 Srt=— 3 SU, 
and substituting for ¥7* or {U’, its value found in the 
last article, we obtain 
D,=2-5°QE. 
37. The expression for r* (Art. 31) may be conveniently 
written thus : 
7*= ’ [(44000) + 4§ (48001) + (41008) } + 6(42002) 
+ 8(88101) + 44 (82102) + (32012) } + 4(22211)] 
+ 23'(2222) + 4.3 (22211) ; 
or, arranging according to the strict descent of powers, 
and substituting for the symmetric functions, %(2222) 
and 5(22211), their values given in Art. 35, 
+= &’§ (44000) + 4(43001) + 6(42002) + 4(41003) 
+ 8(33101) + 4(32102) + 4(32012) + 4(22211) § 
Moreover (Art. 31), 
