ESSAYS 289 



<f> . yjf and (<£)" must be employed to denote algebraic multiplication and involution. 

 I use O" to denote an operator which raises to the «th operative power — so that 

 |"n»]<£ = [<£]" = <£" ; but this will not be required in the present paper (see end of l2'o). 

 2'0. The following elementary theorems will also serve for examples. 



\piQ^ = [^O r ][/ ? O r ] =pi{p*O r ) r =pvp* r O ri ; 



[piO r T =P 7 p qr P qri • • .p qr ' 1 ' 1 ■ O r "=^« (rn - 1,/<r - 1) . O'" 2-1 



when n is a positive integer. If m and n are positive integers 



[P«O r T [p«O r T = [piir m -l)Hr-l) . r m Jp q[ r n -l W -l) , Q r n ] = [^? r f + " J 



from which 2"i may be proved to hold for all real values of n. 

 As corollaries, 



[^O r ]" 1 = Vo\p' 1 ; [^90 r ] 1 /"»=^ 8( *- 1,/( * - y . O r where ssa^/r; 



[0] n = [-0] n = [0 _1 ] n = [-0 _1 ]' , = 0, if n is an even integer ; 



[0]" = 0, [-0]"= -O, [0 _1 ] n = _1 , [-0 _1 ] n = -O"* 1 , if« is an odd integer. 2'2 



The operative roots of these expressions involve algebraic roots of unity. 

 It can easily be shown by substitutions that, whatever n may be, 



[a + 0]* = na + and [j^J =-^ ftQ 23 



If* and (j) are two operations Bo + Co t + Do 3 + . . . and bO + cO 1 + do 3 + . . ., 

 then by substitutions or by Maclaurin's Theorem, 



$$ b [BO + CO 1 + DO 3 + . . .][bO + cO z + do 3 + ...] = BbO + {Cb* + Bc}0* + 

 + {Db 3 + 2Cbc + Bd} O 3 + {Eb i + 2 l Db*c+C(2bd+c i ) + Be}O i +{Fb t > + 4Ebc + 

 + 3D(b*d + be*) + 2C(be + cd) + B/}O s + {Gb° + sFPc + 2E(2b 3 d + 3 £V») + 

 + £K36'e + tbed + c*) + C{2bf + zee + d») + Bg)0* + {/fb 7 + 6Gb>c + 

 + $F(b*d + 2b 3 c*) + 4E(b*e + 2>Pcd + be 3 ) + 2>D{b l f + 2bce + bd* + c*d) + 

 + 2C(bg-+cf+de) + BA}0 1 + e\:c\ 2-4 



The analogy between this " operative multiplication '' and ordinary algebraic 

 multiplication of polynomials may be examined by the reader. It will be observed 

 on trial that the expansion is much more cumbersome if the subject operation <p 

 has an absolute term a. 



3*0. We now proceed to find formulae for the operative involution of multi- 

 nomials. The first case is easy. 



[a + bO]* = a + b(a + bO); 



[a + bO] n =a+ab + ab' + . . . + ab n ~ 1 + b n O = a(i -b n )l(i -b) + b n o ; yi 



which holds for all real numerical values of n. 



Attempting similar substitutions for the general case, 



[a + bO + cO* + dO*. . .]*= {a + ba + cat + da 3 . . .} + {b* + 2cab+' 3 da i b . . .}0+ {bc + 

 + c(b t + 2ac;) + d(3ab* + 3a*c)...}O i +{c(2bc=2ad) + bd. . .}O s + {c 3 . . .}0* + etc; 



[a + bO + cO^dO 3 . . .] 3 ={i+b + b*}a+{i + 3b + b i }ca*+{(i+b)2C 2 + (i+ 4 b + 3 b* + 

 + 6*)d}a* + {(l+4£>? + (5 + 6£)rt/+<:*}a 4 -4-. . . + terms in O. y 2 



This is already too involved, and becomes unworkable for higher values of n. 

 But the problem will be solved in a certain case by another method in 6'o, and it 

 will be shown that the first term of the expansion (free of O) is the algebraic 



