59 



Note Relative to Peirce's " Linear Associative Algebra." James Byrxie 

 Shaw, D. Sc. 



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

 soni.- 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 '' 2, p. 4, 

 and the last three lines of page H'J. The following method of exemplifying the 

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

 ylgabra of this book is an operator of a niatrical kind upon a ground of what we 

 iii.iy call vectors. The whole work is tlius 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 forms. 

 Thus we may write, (a, .i, } , being vectors such tkat S. a /3 y = 1, and tj, I2, I3, U, 

 being quaternions such that S. l^ A. l^ 13!^ = 1 *). 



=: aS.^)(). 



= aS.}«(). 



= aS.3) +/3S.}a(); j = aS.ya{). 



= «S.,?>(); j = aS.ya{). 



= aS.>'a()+/3S.a,3(); j = aS./3y(). 



= aS./3>() + /3S.>'a() + yS.a/?();j = aS.ya() + /?S.a5();. 



k = aS.a{i(). 

 a%, (■. = aS.,3y()+/3S.ya(); j = aS.ya{); k=raS.aii{). 

 a'%, i = -l,S.{)A. 1,1 J, - ^4 S. ( ) A. hl,l, ; j = - I, S. {) A. h 



IJ,; ^-^-/jS.O A. /,/,,%. 

 ba, i = - /x, S. 1 ) A. ^3 /, l,+l,S.{) A. /, I, /, - /, S. { ) A. /, h h ; 



j = hS.{) A. IJ,L_ - h S-( )A. hU_h \k = -h S.( )\.hlJ,. 

 b^, i = - /i S. A. I,UI, + /, S. ( ) A. lj,h_ ; J= /i S. ( ) A. IJJ. ; 



k = — bj, S. ( ) A. l,loj3 + US-' ) A-Uhl-i- 

 c„ i= -l,S.( )A.l,hh+l,S.()X.UJ2; ./ = /, S.( )A./, ?, /, ; 



k = —aliS.( ) A./3/4/1— ;iS.(')A./i/J3 + /4S.()A./,/i/,. 

 d,, i= ,3S.al3{ ); j^aii.a3{); .(—aS. )«(). 

 63, i = — /i S.( ) A././,/, ;./ = — /, S.( )A. ;./,/, -rkS.OA.l.lJ,; 



k = l,S.{ ) A./,/,/,— /., S. ( ) A. 1,1 J,. 



* A. IM* -= S. V^jV/jVi,- V/,.V.V'3V;,— V,'3.V.VJ,V^-V«. V.V/,V?3 

 S.liA.lj3U = — S. L A. l^l^l, =-.'-'. /- A.?J,^ = -.«?, A. IJJ:-.. 



