242 Mr. DE morgan, ON TRIPLE ALGEBRA. 



on which it is only to be observed that any powers, or products, or products of powers, of solutions, 

 are themselves solutions. The most convenient modulus is that which, in one or more definite 

 cases, reduces the system to the simple single or double Algebra already in use. In this common 

 Algebra, in its widest form, there are two unit-symbols, say ^ and ri, usually (not necessarily) 

 representing units of length taken off on the rectangular axes of w and y ; and the laws of com- 

 bination are p = ^, rf = - f, ^»j = »j^ = ri, which give '^rf = - f = (?»;)»;, &c. The modulus of 

 multiplication of af + hn is ■\/{a? + b-). Sir William Hamilton seems to have passed over 

 triple Algebra altogether on the supposition that the modulus, if any, of a^ + btj +c^ must be 

 ^/(a' + 6^ + c'). It is certain* that there cannot be a system of triple Algebra with such a 

 modulus ; but it is by no means requisite that the modulus should be a symmetrical function of 

 a, b, and c. I should also notice that in Sir W. Hamilton's quadruple Algebra there is a complete 

 departure from the ordinary symbolical rules : AB and BA have different meanings. 



§ 2. One mode of derivation of systems of triple Algebra. Let a^, bij, e^, represent lines of 

 o, 6, and c units measured on the axes of x, y and s. Let it be a condition that 6=0, c = 0, 

 reduces the Algebra to the common single system ; which might be worded thus : let the Algebra 

 of the axis of x be the common single Algebra of positive and negative quantities. Also let ri and T 

 be interchangeable, and related in the same manner to ^. We have then, for the forms which 

 define the actions of the unit-symbols on each other, 



p means p ij^ means p^ -Y q>i + qX,i 



rf o^ + 6>; + c^, X,i, l^ + mri + nX^^ 



^' o^ + cij + 6^. fv l^ + n>i + m^; 



audit will be found upon examination that the equations p») = ^(^>;)> ^1' = li^l)^ '''^ = '?(''D' 

 vV = Ult)^ r? = UW' ?f = HW' UlO = l(tO = t(U), ^vill be .satisfied by the following 

 conditions ; in using which care must be taken not to form new ones by introduction of subse- 

 quently vanishing factors without recurring to the original forms. Some of these conditions are 

 included in the others, but it is nevertheless desirable to be reminded of them. 



(1.) a(q - c) + p{q - 6) = /(« - p)- (4.) l(m + n) = 0. 



(2.) l" + mp + na = a + (h + c)l. (5.) ■■2mn = m. 



(3.) f + ma + np = p + 2ql. (6.) jn^ + n^ = «. 



(7,8.) In = (q - b)m = (c - q)m. (11.) {q + c) (q - c) = am - pn. 



(P, 10.) Im = (f/ - c)iH = (6 - q)m. (12.) (<7 + c) (q - b) = an - piii. 



From (5.) and (6.) we have either 



2' "-4' '" = -i' »* = i- 



Proceeding by analogy, we might expect the triple Algebra which is the proper extension of 

 the common double one to give »/ = - ^, 'C ~ ~ ^^ ^^^ necessary conditions of which are 



(13.) a I -i- ab + cp = - 1. (M.) aw + ft- + cq = 0. (15.) am + be + cq = 0. 



• Any one who will try to get three squares in which accented 

 and unaccented letters enter symmetrically, and of which the sura 

 is equal to the product of a^+b^ + c^ and a'- + 6'^ + c'^ is engnged, 

 whether he know it or not, upon the following problem ; — To tind 



three points of a sphere, each of which is opposite to both of the 

 other two; also three otlier points each distant by a quadrant from 

 each of the first three. 



