892 PROCEEDINGS OP THE AMERICAN ACADEMY 



XIV. 



ON THE APPLICATION OF LOGICAL ANALYSIS TO 

 MULTIPLE ALGEBRA. 



By C. S. Peiece. 

 Presented, May 11, 1875. 



The letters of an algebra express the relation of the product to the 

 multiplicand. Thus, iA expresses the quantity which is related to A 

 in the manner denoted by i. This being the conception of these alge- 

 bras, for each of them we may imagine another "absolute" algebra, as 

 we may call it, which shall contain letters which can only be products 

 and multiplicands, not multipliers. Let the general expression of the 

 absolute algebra be al -\- bJ-\- cK -j- dL -\- etc. Multiply this by 

 any letter i of the relative algebra, and denote the product by 



( Jjffl -|- A.p -[- ^-f + etc.) 1. 

 -f {Ba 4- m 4- B^c 4- etc.) /. 

 -)- etc. 



Now we may obviously enlarge the given relative algebra, so that 



i = A^ ^^^ -|- A., ^^„ -\- A.. i\„ -\- etc. 



+ A ^2. + ^2 hz + -^J '23 + t;tC. 



-\- etc. 



where i^^ i^^ etc. are such, that the product of either of them into any 

 letter of the absolute algebra shall equal some letter of that algebra. 

 That there is no self-contradiction involved in this supposition seems 

 axiomatic. 



In this way each letter of the given algebra is resolved into a sum 

 of ternas of the form a A : B, a being a scalar, and A : B such that 



(A:B) {B:G)= A:G. 

 (A:B) (C:B)=:0. 



The actual resolution is usually performed with ease, but in some 

 cases a good deal of ingenuity is required. I iiave not found tlie pro- 

 cess facilitated by any general rules. I have actually resolved all the 

 Double, Triple, and Quadruple algebras, and all the Quintuple ones, 

 that appeared to present any difficulty. I give a few examples. 



