# Perpetuant sentence example

perpetuant

- If 0=2, every form is obviously a ground form or perpetuant, and the series of forms is denoted by (2), (22), (23),...(2K+1)....
- Similarly, if 0 =3, every form (3K+12,x) is a perpetuant.
- Therefore every form of degree 2, except of course that one whose weight is zero, is a perpetuant.
- Hence every product of A 1, A2, A3, A4, which contains the product A 4 A 3 disappears before reduction; this means that every seminvariant, whose partition contains the parts 4, 3, is a perpetuant.
- When 0=0' =O, We Have The General Perpetuant Of Degrees I, I.Advertisement
- For the case 8=1, 0' =2, the condition is a i r 1 72 = A032=0; and the simplest perpetuant, derived directly from the product A 1 B 21 is (I)a(2)b-(21)b; the remainder of those enumerated by z3 I z.
- To represent the simplest perpetuant, of weight 7, we may take as base either A2B 1 B 2 or A l A 2 B2, and since Ai+Bi =o the former is equivalent to A 2 ArB 2 and the latter to A 2 B i B2; so that we have, (1 -f-aix) (1 + a2x).
- By the rules adopted we take A?B 2 B 3, which gives (12)a(32)b - (1)a(321)b+ao(3212)b, the simplest perpetuant of weight 7; and thence the general form enumerated by the generating function 1 -z.1-z2.1 - z3 ?