XVIII. On the Foundation of Algebra, No. IV., on Triple Algebra. By Augustus 

 De Morgan, V.P.R.A.S., F.C.P.S., of Trinitij College; Professor of Mathe- 

 matics in University Colk'ge, London. 



[Read, October 28, 1844.] 



In the Philosophical Magazine for July 1844, Sir William Rowan Hamilton lias published the 

 first part of a paper read before the Royal Irish Academy in November 1843, headed 'On Qua- 

 ternions, or on a new System of Imaginaries in Algebra.' To this paper I am indebted for the idea 

 of inventing a distinct system of unit-symbols, and investigating or assigning relations which define 

 their mode of action on each other. The systems which i shall examine differ entirely from that of 

 Sir William Hamilton, both as being triple instead of quadruple, and as preserving, in their laws 

 of operation, a greater resemblance to those of ordinary Algebra. 



§ 1. Description of triple systems. A system of Algebra of the n"' character is one in 

 which there are n distinct symbols, ^|, ^o, ... ^„, each of which is a unit of its l<ind, of a difference 

 from all other i<inds such that "ifi + cf,^2+ ... cannot be equivalent to 6,^, + 62^2+ ... unless 

 a, = 6,, Oa = 62, &c. This condition however is connected with the interpretation: a perfect sym- 

 bolical system might very well exist without it. Having assumed a system, and also the ordinary 

 laws of addition and subtraction, the introduction of the operation of multiplication requires that 

 meanings should be assigned to ^1^2, ^1^3, &c., so that each of them may be regarded as coincident 



with such a form as a,^, + a.,^,, + On the manner of assigning this form the properties of the 



system entirely depend ; and if we are to preserve the ordinary rule of the convertibility of multipli- 

 cations and divisions, we must not only provide that ^, f^ = f^f,. Sic, but also that ^,'^2^? ... shall 

 give the same result in whatever order the operations are performed. This role relative to mul- 

 tiplication may be reduced to two simple rules, AB = BA, and A{BC) = {AB)C. It is exactly 

 the same thing as to additions, the convertibility of which is contained in the rules A + B = B + A 

 and {A + B) + C = A Jr (B + C). This second rule is generally concealed in the common rule 

 of signs, according to which A + {B + O or A + [0 + B + C) is, by the assumed distributive 

 character of the sign +, allowed to be transformed into ^ -t- (+ S) + (+ C) which again by the 

 rule of like signs, becomes A + B + C, a symbol identical in meaning with (A + B) + C- We 

 might also use the signs x and — in the same absolute manner, and assume a corresponding dis- 

 tributive character, and rule of like and unlike signs: considering x a and -H a as abbreviations of 

 I X a. and 1 -f- a. But it will be enough for my present purpose to note that the complete conver- 

 tibility of multiplications will be secured if every triple combination, as ^ifj^si ^??v!! ^^- ^^^ 'i 

 meaning which is independent of the order of the operations. 



Having settled the system, it must next be inquired, for the sake of the interpretation, what is 

 the modulus of multiplication, namely, what function of a,, a^, &c. is it which, in the product, 

 has the same value as the product of the functions of the factors. If, agreeably to the laws of 

 tlie system, the product of a,^, -1- a,lc,s+ ••■ and ft',^, + n'.>'^.,+ ... be ^,5, + y/af. + ..., A^, A.., &c. 

 being definite functions of a,, o',, a.^, a\, Sic, the modulus is to be found frym the solution of 

 the functional equation 



</»(«,. ",...) X (•/) (a',, a',, ...) = (p(/l,. A,,...), 

 Vol. VIII. I'aut III. I i 



