26 The Rev. C. Graves on a system of Triple Algebra. 



pable of being readily put, in evidence, because the expression 

 vanishes for 5= + 1. In other words, as 5 partakes of the cha- 

 racter of + 1, these two factors are of such a nature that either 

 of them, like zero, assimilates the product into which it enters 

 to itself. If both enter into the same product, they must make 

 it like their own product 1—5^(1), which, by the definition of 

 5, is equal to zero. 



11. The same kind of reasoning admits of a more interest- 

 ing application to the problem of finding the moduli of multi- 

 plication in the triple algebra whose units are 1, s, and «; or 

 in that other whose units are I, m, and w. 



As the operation n is defined by the equation 



^1-5(1)) 

 ''" a/2 ' 

 it appears that, when s degenerates into + 1, « becomes ; and 

 when s is equal to — I, n is equal to + -/ — 2. In the former 

 case the equation (2.) is reduced to 



(.r +3/) (.2'' -f- y ) = x" +y' • 



In the latter we shall have 

 {x-y± \/^^.z){a,'-i/'±\/^^.z') = x"-y"± x^'^.z". 



Thus we obtain the two real moduli of multiplication of the 

 system whose units are 1, 5, and n. 



12. Let us next pass to the consideration of the triple sy- 

 stem, whose three units are /, m, and «, as defined by the equa- 

 tions (4<.). 



According as 5= +1, or —1, these latter become 



/= v/2, 7rt = 0, n = 0; 



or 



/=0, m= s/% n—± a/— 2. 



And, if we introduce these two systems of vahies successively 

 into the equation 



(/? + mij + «^)(/r + mii + vXl) = /r' + mil' + «$", 

 we shall derive from it the following : 



__ >/2.fr=^", 



a/2(>)± v/^.?)(»)'± ^-1.^') = V'± 'vZ-U", 

 which furnish the modular relations belonging to this system 

 of triple algebra. 

 DiibIin,Jan. 12, 1849. 



[To be continued.] 



