204 REY. R. HARLEY ON SYMMETRIC PRODUCTS, AND 
+ 12(51433) + 16(51343) + 6(51334)} + $16(54322) 
+ 4(54252) + 4(54223) + 4(53422) + 16(53242) 
+ 4(53224) + 4(52432) + 16(52423) + 4(52342) 
+ 4(52324) + 4(52243) + 16(52234)} + 6§ (53332) 
+ (53323) + (53233) + (52335) } + 24(44440) 
+44 (44431) + (44413) + (44341) + (44148)? 
+ 12§ (44422) + (44242)? + $10(44332) + 8(44323) 
+ 10(442383) + 10(43432) + 10(43423) + 8(42433) 
+ 4(43333)] —8-5QEU*. 
Applying the theorem in Art. 29 we have, 
LUt=23'(844) + 4.3)(83311) + 125'(82222) + 43)(7531) 
+ 43'(7432) + 163 (74221) + 16373321) 
+ 123)(6622) + 4.5'(6541) + 203'(65311) 
+2435 (65221) + 123'(6442) + 203'(64411) 
+ 243'(6433) + 283'(64321) + 523'(63322) 
+ 24.3; (5542) + 245/(55411) + 483'(5533) 
+ 163 (55321) + 723)(55222) + 445'(54421) 
+ 683'(54331) + 483 (54322) + 363 (53332) 
+ 144.3'(4444) + 245'(44431) + 723) (44.422) 
+ 563 (44332) + 243'(43333) — 8-5QES,U”. 
But 
2 (844) =- 8-5°Q?E?, 3(83311)=11-5°Q?E?, 3(82222) 
=3-5°Q'E’, 3(7531)=3-5°Q?E?, 3 (7432) =4-5°Q7E’, 
3(74221)=6-5°Q?E*, 3(73321)=6-5°Q?E*, 3 (6622) 
=11-5°Q?E?, 3 (6541) =-9-5°Q?E®, 3'(65311)=7- POE’, 
2 (65221) =-3-5°Q@?*E®, 3 (6442) =—- 10-5°Q’E?, 364411) 
=2-5°Q°E’, 3(6433)=6-5°Q@’ EH, 3(64321) =6-5°Q°E* 
3 (63322) =3-5°Q2E?, 3 (5542) =8-5°Q?E’, 355411) 
=-5°Q?E’, 3 (5533) =0, 3(55321)=-6-5°Q?E’, 3 (55222) 
=3-5°Q?H, 3(54421)=0, 3(54331)=6-5°Q?E? 
(54322) =-3-5°Q?H?, 3'(53332)=0, 3'(4444) =2-5°Q?E?, 
3 (44431) =— 2-5°QPR*, 3/(44422)=5°Q2E*, 3/(44332)=0, 
(43333) =0, and (Art. 35) 2U?=- 4.5°QE. 
Consequently, 
DUt= 4. 5°Q?E? ; 
