124? The Rev. C. Graves on a system of Triple Algebra, 



the first and second cases, a factor actually vanishes ; but not 

 so in the third and fourth. In these, two real moduli, of dif- 

 ferent kinds and belon<Ting to different factors, becoming equal 

 to zero, annihilate the two real moduli of the product, and so 

 cause itself to vanish. 



So again in the case of the trinomial x -\- s{y) -\- n[z). It 

 appears from the equations (3.) that it has two real moduli of 

 multiplication, x+y^ and {x—y)^ + 2z^. That is to say, if 

 {x + s{y) + n{z))[x' + s{y')-^n[z')) = x''^s{y")+n{z")', 



we shall have 



{x-Yy){pi^+i/)=x"+y\ 

 and 



(J^x-yf + 25;2) ( [x^ -y'f + 2z'^) = {x" -y"f + 22"\ 



The factor x-{-s{y) + fi{z) vanishes if both its real moduli vanish, 

 but not otherwise. And the product x" + s{y") + n{z") will not 

 vanish unless both its real moduli likewise vanish. But this 

 may happen in different ways. 



1. We may have x+y = 0, and {x—i/)^-\- 22^ = 0; condi- 

 tions equivalent to x=y = 2 = 0, and therefore involving the 

 annihilation of a factor. 



2. x'+y' = 0, and {x' —y'f + 22'^=i0; conditions which, in 

 like manner, annihilate the other factor. 



3. x + y = 0, and {x^—y'f + 2z'^=0; equivalent to a'+^=0, 

 x'—y'=0, and 2;' = 0. 



4. {x— y)'^ -{-22^=0, and x'+y' = 0; equivalentto .r— 3/=0, 

 2=0, and x'-\-y'=0. 



Here, as before, we see that the complete annihilation of 

 the product may be brought about either by the complete 

 annihilation of either factor, or by the vanishing of two mo- 

 duli, of different kinds, occurring one in each of the two 

 factors. 



Similar observations apply to the system of triple algebra 

 discussed by Professor De Morgan in the Transactions of 

 the Cambridge Philosophical Society, vol. viii. ; and to the 

 closely related system, of which 1 have given a brief account 

 in the Proceedings of the Royal Irish Academy, vol. iii. pp. 5i 

 and 57. 



In both the systems just referred to there are two real mo- 

 duli of multiplication ; and in both a product of two factors 

 may vanish without the vanishing of either of the factors them- 

 selves. 



9. It is not only when the two factors 1 +s(l) and 1— s(l) 

 come together that we meet with a singular result. The 

 occurrence of either of them in a product affects it in a re- 

 markable manner. 



