59 



NoTK Relative to Peirce's " Linear Associative Algebra." James Byrnik 

 Shaw, D. Sc. 



I have no doubt many readers of Benjamin Peircc's classic work have found 

 somt! difficulty in its perusal from the lack of examples of the algebras de- 

 veloped. That such a completion of the work was intended is shown by 1[ 2, p. 4, 

 and the last three lines of page 119. The following method of exemplifying the 

 subject may be of use or helj). It is in a succinct form thus: Every unit in an 

 i'lgabra of this book is an operator of a matrical kind upon aground of what we 

 may call vectors. The whole work is thus a treatise on groups of such operators. 

 This explains its abstruseness. Now for all cases in which the ground consists of 

 two or three or four vectors, the units can be represented by the linear vector 

 operators of quaternions, or linear quaternion operators. The relative forms given 

 by Mr. C. S. Peirce may be immediately translated into such quaternion lorms. 

 Thus we may write, (a, fi, y, being vectors such that S. a fi y = 1, and Zj, l.^, l-s, U, 

 being quaternions such that S. /j A.ij '3 ^4 = 1 *)• 

 Algebra ai, i = aS.(iy{). 

 " bi, (' = a S. )' « ( ). 



a,, i = aH.ihi) + (iS.ya{)', j=:aS.7a{). 

 h„ i = aS./3>'(); j = aS.ya(). 

 " C2, i = a8.ya()+/3S.«/?(); j = aS./?y(). 

 " d^, i^ hS.{) A.l.lJ,; J = kS.{)A.lJ,l,. 



" a3, t = aS./3>() + /3S.>'a()+7S.a/?();j=aS.>'a() + /3S.a^(); 



k = aS.afi{). 

 a'„ i = «S./?7()+/3S.y«(); i=^aS.ya(); k = aS.a(i{). 

 a-3, i = -l,S.()A. 1,1 J ^ -l^S.{)A. hhk ; j = - I, S. ( ) A.l, 



IJr, k^-l,8.()A.l,l,l,. 

 h„ i =~l„S.{)A.l,Ul,+l,S.{)A.lJ,l,-l,8.{)A.l,LI,; 

 j = l, 8.{)A.l,l,l, — l^B.{ )A.IJJ,; k = -l,S.()A.l,l,l,. 

 b'3, i = - /i S. ( ) A. /,,/,/, + /, S. ( ) A. I J J., ; J= /, S. ( ) A. I J, I, ; 



k = — h,l, S. ( )A.lJJ,-}-l^S.{ )A.lJ,l,. 

 c„ i= -/, S. ( ) A.lJ,l,+l^f^.() A.l,l,L; .;■ = /, S.( ) A./, /,/.,; 

 A = - a /, S. ( ) A. /3 V, — /, S.( )A. /,/o/3 + /^ S. ( ) A. I J, I,. 

 " dj, i= ii^.afii }; j^n^.n.f3(); k=aS.ya(). 



63, 1 =-/,8.( )A.l,!J,-j^-l,^.( )A.I l,l,+l,S.{)A.l,IJ,; 

 ifc = /,S. ( )A.IJJ^ — U^ S. ( )A.l,tJ,. 



*) A. l^lal^ = S. V/^V/^VJ, — V^j.V.V/jVi,— V/^.V.M.V/j— VJ, V.V/^V^a. 

 S. liA.lj3li== — S. L_ A. IJJ, -H. l^ .\.l,l,L_ = - 8 ?, A. IJJ.,. 



