THEIR APPLICATION TO THE SOLUTION OF EQUATIONS. 201 
3 (43221) =0, 3(42222)=0, 3(333821)=0, and 
> (33222) =0. 
Hence 
FP —=645'Q"; 
38. Again (Art. 31):— 
U?= 2’ [4 (42002) + (40220)? + 2§ (83101) + (80311)? 
+ 24 (82210) + (32021) + (81202) + (80122) ?}] 
+23(41111) +23(22211) ; 
or since (Art. 35) 
=(41111) =—3-5QE, and 3(22211)=-—5QE, 
U?=%’5 (42002) + (40220) + 2(33101) + 2(32210) 
+ 2(82021) + 2(81202) + 2(30311) + 2(80122) 
—8-5QE. 
Next, 
U=’$ (21001) + (20110) } = (21001) + (20110) 
+ (12100) + (11020) + (10201) + (10012) 
+ (02011) + (01210) + (01102) + (00121)?. 
Nov, 
Us=>’§ (42002) + (40220) + 2(83101) + 2(82210) 
+ 2(32021) + 2(81202) + 2(380311) + 2(80122)?U 
—8-5QEHU. 
Hence if, in the function affected by 5’, we substitute for 
U its value, develope, reduce, &c., as in the last article, we 
a obtain 
=’ [§ (63003) + (60330) } + § (62112) + (61221)? 
+34 (54102) + (52014) + (51240) + (50421)? 
+ 8§ (53022) + (52320) + 52208) + (50232) } 
+4 (538211) + (52131) + (613812) + (51123)? 
+34 (44310) + (44013) + (41430) + (41403) } 
+ 6§ (44121) + (42411)? + 64 (43302) + (43230) 
+ (42033) + (40323) } + 4§ (43113) + (41331) } 
+ § (43212) + 2(43122) + 2 (42821) + (42231) 
+ 2(42213) + (42123) + (41322) + 2(41232) ? 
+ 8(42222) + 45 (33321) + (83312) + (33231) 
+ (83132) +24 (83222) + (32422)?] -8-5QEU ; 
