932] 
285 
ON SYMMETRIC FUNCTIONS AND SEMINVARIANTS. 
This gives 
Here 
and 
— 3 (CH 2 + HdH, + 6e + bSa. 3 ) Sol 2 
- 2 (CH, + U + bSa 2 ) So? 
+ 5 (CH 4 + 3dH 3 + 6eH 2 + 10fH 1 + 1 og + bSoL 5 ), 
which is found to be 
= -S(BD + C 2 + bSoL 3 ) Sol 2 
- 2 (BC + bSd 2 ) Sol 3 
+ 5 (BF+CE + D 2 + bSa 5 ). 
bSa 3 = - SolSol 3 = - Sol* - Sol 3 /3, = - E - BD, 
bSoi 2 = - SolSol 2 = - Sol 3 - Sa% =-D- BC, 
bSoc 5 = - SolSol 5 = - Sol 6 - Sol 5 (3, =-G-BF; 
the expression thus is 
= - 3 (- E + C 2 ). C that is, - 3 (- Sol 4 + SoPfi 2 ) Sol 2 
— 2 (— D ).D -2 (-Sol 3 ) Sol 3 
+ 5 (— G + CE + D 2 ), + 5 (— Sol 6 + Sol 4 (3' 2 + Sol 3 /3 3 ). 
Sa 2 Sa 4 = Sol 6 + Sol 4 /3 2 , = G + CE, 
Sol 3 Scl 3 — Sol 6 + 2Sa 3 /3 3 , = G + 2D 2 , 
Sol 2 Sol*/3 2 = Si 4 /3 2 + nSa 2 S-T, = CE + SC 3 ; 
- 3 {- G- CE+(CE+SC 2 )} ■ 
— 2 (— G — 2D 2 ) 
+ 5 (— G + CE + B 2 ), 
which is = bCE + 9D 2 — 9C 3 (a non-unitary form). This then should be the value of 
Here 
and the whole is 
\Q — bay, = cdu + 3 dd c + Cyedd + 10+ 1 bgd/ — 5b, 
operating upon 
Sol 3 /3 2 , = CD = of— obe + cd + 2 b 2 d — be 2 . 
31. There is for non-unitariants a theorem which is a much more simple form 
than the transformation of it afterwards obtained for seminvariants: viz. for any 
non-unitariant we have AU = 0 = (9 6 + bd c + eda +...1 U\ attending only to the portion 
of U which is of the highest degree, it is clear that we have (bd c + cd d + ...) U' = 0, 
and if we herein diminish the letters, then (9& + bd c 4- ...) U" = 0, where U" is what 
^ becomes by a diminution of^ the letters; that is, U" is a non-unitariant, viz. in 
any seminvariant, the terms of highest degree U' are obtained from a non-unitariant 
U' by a mere augmentation of the letters: e.g. 2e — 2bd + c 2 is a non-unitariant 
weight 4; augmenting the letters, we have 2bf—2ce + d 2 which with a change of 
sign is the portion of highest degree of the non-unitariant 2g — 2bf+ 2ce — d 2 .