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[ 13 ] 



II. 



ON THE PLACE OE THE AUSBERNUNGSLEERE IE 

 THE GEXERAL ASSOCIATIYE ALGEBRA OE THE 

 QUATEEXIOX TYPE. By CHAELES J. JOLY, M.A., 

 P.T.C.D. 



[Read, Februart 13, 1900.] 



Theee is a cardinal distinction between Quaternions and other alge- 

 braic systems of space analysis. So far as I know Qnaternions and 

 algebras of the Quaternion type alone are both. Associative and 

 Distributive. Of the other systems, some are associative but not 

 distributive, some distributive but not associative. The Ausclehn- 

 ungslelire is distributive but only partially associative. 



It is worth while inquiring -whether the Ausdehnungslehre can be 

 included in the distributive and associative algebra whose units obey 

 the laws of Quaternions 



?.s" = - 1 and i^^it + ifi, = 0. (a) 



"We shall, in what follows, use the term vector (when not otherwise 

 qualified) to denote a linear function of some or all of the ixnits with 

 scalar coefficients, and we shall generally employ small Greek letters 

 for the symbols of vectors. Also we shall restrict the words j)roduct^ 

 multiply^ &c. to the results of operations and to operations in accord- 

 ance with the laws of the units. Eurther, when we speak of a set of 

 units, we imply that they satisfy equations (a). 



If in accordance with (a) we form the complete product or simply 

 \hQ ]^ro(liLct of any number of vectors -z<rj, tsto,^ . . . -zt,, and reduce as far 

 as possible by the aid of the fundamental relations, it is obvious that 

 the result will, in general, consist of sums of irreducible products of 

 the units of the orders n, n - 2, n - 4, &c., each product being 

 multiplied by a scalar coefficient. Hence we may write if we separate 

 these sums of products into groups of the same order, 



TtTlTSTz'^Z . . . "STn — 'n'^l'^2 • • • "^n "I" ' >;-2'^l'^2 • • • "^n "^ &C. 

 = 2 yn-2m'^l'^2 • • • '^M 



^u-tm being a sign of selection of groups of products of the units of 

 the order indicated by the suffix. 



