196 REV. R. HARLEY ON SYMMETRIC PRODUCTS, AND 
Or, working with U, in place of 7’, 
UHL a7 (ty 5 +23 2s) 5 
fe 6S SS (L421) Sat aap 2 at ay i), 
which confirms the value of =7?._ Consequeutly, 
Dj=52r=5>0=0: 
31. Again: — By the commutative property established 
in Art. 19, 
T=! (aj ay + 2a} a X5)7; 
and developing 7 according to descending powers of 2, we 
have 
T= LF 02 + 27} ay Hs +22 02+ 2H, £3 y+ 2a, 2, 22+23 22 
Fatty 3 Vy + U5 UE + 2lks a Ws + HG US 5 
whence, multiplying by vj 23+ 2¢} 2, 2;, reducing by means 
of such relations as (Art. 19) 
Dv} Hg = DU} ty 3, BUX, UX, W{= DU} Wy L, X;, Se. 
and collecting similar terms, we find 
TA fa a+ Aa WL; + Uj Ly U5) + Oa ay v5 + BA} X3 as Ws 
+ 4a} a9 a3 U5 + a @3 Ly U5) + Qay w3 3 wy + Bay Ly H3 &y He 
+ ay U2 %5 Ui 25% 5 
or, since 
Sw R= riage 7, end 
Dy! (Way W3 Wy Lz + Uj Hy Ly i Ls) = ZL, Vy XZA ys, 
r= 5 Sat af + 4(v} v3 v5 + x} vy v3) + Cat x5 03 + Bait x x, 5 
+ 4x} ay a3 15+ aX] a3 Wy V5) + Ay Was ay Ws$ 
+ WBA} Wy Us WEA ADA ay 5 Ws Ws 5 
o. Jt=85 24 8445 7 vi 7, +632} 23 224 8d ai avs, 
HAS a} Uy 03 y+ 12D ay wy V5 H+ BOR Xi Hwy 4 25. 
Or, working with U, 
W=3) (sie, 4-47 2 7), 
developing U, multiplying, reducing, &c., there results 
USB § (wh 08 a2 + af a8 03) + Qa ay my yg + (ah ah ya 
+ ata ay a) +2 (ah ahaha, + wh ad aia + ah ay a2 a8 
+ 23 ay 0} 02) +2 (xt #3 03 a, y+ Ui Le Hs UEH5)t 
= D'S (wt av} 2 + vf x a}) +2 (ah 29 ary 5+ vf a9 as 5) 
