202 REV. R. HARLEY ON SYMMETRIC PRODUCTS, AND 
*, SU? =23(633) + 25 (62211) +33(5421) +62(5322) 
+83(53211) + 63(4431) + 123(44211) 
+123(4332) + 83(43311) + 63(43221) 
+483 (42222) + 24:3 (33321) + 123(83222) 
-—8-5QEXU. 
But 5(62211)=0, 3(5322) =0, 3(53211)=0, =(4431) =0, 
and SU=0; hence making these substitutions, and also 
for the other symmetric functions whose values are given 
in the last article, we obtain 
S'=6.500%, 
which confirms the value of =r°. 
39. By Art. 82, 
D =-137°= Osa 3 sys . 
3 3 3 3 ? 
and substituting for 27° or >U? its value (Art. 37 and 38), 
we have 
Dee. 
40. Since (42222) = 2(42222) = E’[2] =0, and 
&’§ (83222) + (82822)} = 3(33222) = EH’ 3(11) = 0, the 
value of 7* (Art. 37) may be expressed thus : 
7°= >’ [ (66000) + 6§ (65001) + (61005) } + 15464002) 
+ (62004) } + 20(63003) + 24(55101) + $36(54102) 
+ 6(54012) + 6(52104) + 36(52014)} + 244 (53103) 
+ (58018)? + § (44220) + 28 (44202) + (44022) ¢ 
+ 420(44211) + 2(44121) + 20(44112? + 8 (43203) 
+ (43028) } + 40(43113) + $4(43221) + 24(43212) 
+12(43122) + 12(42213) + 24(42123) + 4(4] 223)? 
+ 16§ (33321) + (83312) } + 44(33222)]. 
We might now arrange these functions according to de- 
sceuding powers, multiply by 7°, and proceed precisely 
as in Art. 37, but it will be more convenient to work 
with U. 
41. Since 3 (42222)=0, 3”} (88321) + (83312) + (33231) 
+ (83182)t = 3'(83321)=0, and 3'{(33222) + (82322)t 
= (33222) =0, we have (Art. 38) 
