THEIR APPLICATION TO THE SOLUTION OF EQUATIONS. 205 
and, therefore, (Art. 32) 
D,=}(2D}- 364) =4 (2D}- 5'3U") =} (2D3— 4. 5°@EY ; 
or, since (Art. 36) 
D,=2-5°QH, 
D,=5"Q2E?. 
42. Since 5’ (82222) = 3'(82222)=3-5°Q2EY, 3” (63322) 
4+ (63232) + (63223) + (62332) -+ (62323) + (62233)? 
= 5 (63322) =3-5°Q?E, 3’ § (55222) + (52522)? = 3 (55222) 
=3-5°Q?E?, 3’§(54331) + (54313) + (54133) + (53481) 
+ (53413) + (53341) + (53314) + (53143) + (53134) 
+ (51433) + (51343) + (51334)? = 354331) =6.5°Q2E?, 
S §{ (54322) + (54232) + (54223) + (53422) + (53242) 
4 (53224) + (52432) + (52423) + (52342) + (52324) 
4 (52243) + (52234)? = 3 (54322) —- 3. 5°Q2EX, 3§ (53332) 
+ (53328) + (53233) + (52333)}—=3(53332)—0, 3’ (44440) 
= 3 (4444) =2-5°Q2EY, 3’ (44431) + (44413) + (44341) 
+ (44148) = 3'(44431) =— 2. 5°Q@2E2, 35 (44422) 
+ (44242)} = 5 (44422) = 5°22, S’f (44332) + (44323) 
4+ (44233) + (43432) + (43423) + (42433)? = (44332) =0, 
and 3’ (43333) = 3(43333)=0, we have by substitution in 
the value of U* (Art. 41), 
Ut= > [f (84004) + (80440) } + 2§ (83113) + (81331)? 
+44 (75103) + (73015) + (71350) + (70531)? 
+4§ (74023) + (73204) + (72430) + (70342)? 
4 8§ (74212) + (72241) + (72124) + (71422)? 
+8{ (73321) + (73182) + (72313) + (71233)? 
+ 6§ (66202) + (60622) +44 (65014) + (64105) 
+ (61540) + (60451)? + 10§ (65311) + (63151) 
+ (61513) + 2 senha 65122) + (62521) 
+ (62215) + (61252)? + 6§ (64420) + (64042) 
+ (62404) + (60244)? + 10§ (64114) + (61441)? 
+ 12§ (64303) + (63340) + (63034) + (60433)? 
+ {20(64231) + 2(64213) + 6(64123) + 20(63412) 
+ 6(63214) + 2(63124) + 6(62431) + 2(62341) 
+ 20(62143) + 2(61432) + 6(61342) + 20(61324)? 
