Mb. DE morgan, ON TRIPLE ALGEBRA. 253 



The imperfection of this system, as in the former case, consists in the want of the equation 



A{BC) = iJB)C. 



There is a remarkable new consideration, which presents itself in tliese systems of inverted 



multiplication, as we might call them. When ^ is an inoperative symbol, that is, when ^tj means 



7} and ^^ means ^, the abstract number of common arithmetic, m, may be represented by a line 



^ + Otj + 0^. But, in the case before us, the multiplier tn and the multiplier m^ are very distinct 



things. The former has only the effect of multiplying the length by m, without altering angles. 



But there is still a line which has the effect of the abstract multiplier »», upon a^ + btj + c^: 



it is 



,, c — b b — c ^ 



my + m n + m T. 



a a * 



The product of these two lines is m a^ + mbr] + met- Now the second line represents a line of 

 the length m, on the axis of a: ; not having its subsidiary point at its extremity, but at a finite 

 distance on the neutral plane. And thus it appears that every such lino of the form ^ + 1^ t] - pX. 

 plays the part of the abstract multiplier I to every line of the form n^ + &>; + (?< + (ip)X,- 



^ 8. On looking back to ^2, we see under case A, a perfect cubic form with the equations 

 of signification 



r=6 'i'=i K'-b r,x,-i, r? = -e. e'? = -e- 



Accordingly every product is of the form m^, or according to our usual interpretation, must 

 be laid down on the axis of w. Look at the quadratic and cubic cases that come under C and D, 

 and it will be equally apparent that all products take the form wf -r w(»? + D °'' "'^ + "(i? - X,)i 

 according to the system : consequently all products come into one plane. It would be easy enough 

 to make any number of triple systems, under such a condition. 



The perfect quadratic system under B may be readily developed. Its modulus of multiplica- 

 tion is ^y {(V' + (b - cY \ which will require that, in an explanation resembling that of | 5, the 

 neutral and primary axes should change places. The line m(»7 + ^) is one of no length in such a 

 system, and if w(>/ + ^) be added to a^ + bri + cY, nothing is changed except the position of 

 the imaginary axis. Let all the explanations be as in fi 5, after interchanging the neutral and 

 primary axis : then the system before us is complete when we add to the explanations in S .5, 

 thus altered, the condition that the product of a^ + 6»j + c^ and a'^ + b't] + c'^ is to have the 

 addition (bb' + cc) {rj + ^), giving a certain alteration in its imaginary plane. 



I should have liked to have delayed the present communication until I could have examined 

 these and other cases in more detail. But as, owing to the approach of other occupations, any such 

 delay must have lasted a year, I determined to send my thoughts just as they are, in the hope that 

 others may be induced to pursue the subject. One great point of the interest which attaches to it, 

 is the hope that the generalized notions of interpretation which it gives, will be found applicable to 

 the common double Algebra, which is at present restricted to systems of linear co-ordinates : and as 

 to which, though the restriction is clearly unnecessary, the proper direction of generalization is 

 not seen. 



A. DE MORGAN. 



Univeb«ity College, London, 

 October 9, 1844. 



