120 The Rev. C. Graves ow a system of Triple Algebra^ 



2. It is distributive ; therefore 



s[a) + s{b)=s{a + h)=i{a + b)s{\). 



3. It is a periodic operation of the second^order ; that is to 

 say, s^(a) = a. 



2. The nature of the operation being so far 'defined, let us 

 take two mixed binomials, x-\-s{y) and x' + s{y'), and operate 

 with either upon the other ; for it matters not which is ope- 

 rator and which operand; the result of the operation will be 



{.V + «(j/)) (.r' + s{i/))=xa^ -\ryi/ + s{xy' +ya^) ; 



and putting 



a/' = xoi^ + yy\ and y = xi/ -hyx', 



it is easy to see that we shall have 



{x-1/){x'-7/') = x"-7/>J' ' ' ' ^'^ 



These are the modular equations of multiplication in the sy- 

 stem of double algebra, with which we are at present concerned. 



3. There is no difficulty in assigning a geometrical inter- 

 pretation to the symbol s. 



Draw in a plane the two rect- 

 angular axes of co-ordinates OX 

 and OY, and bisect the angle be- 

 tween them by the right line OA. 

 Then the symbol 5 may be taken 

 to express rotationfrom left toright 

 through an angle of ] 80° round the 

 axis OA. 



Supposing that the real unit 

 be placed upon the axis of ^, 

 it is evident that this geometrical 

 representation fulfils the condi- 

 tions imposed upon s. For, in 

 the first place, 5(1) lies upon the 

 axis of y ; and being at right angles with the real unit, is 

 as much distinct from it as V — I is. Next, this rotation is 

 plainly a distributive operation. Lastly, it complies with the 

 third condition; seeing that a repetition of the rotation through 

 180° brings the real unit back again into its original position. 

 4-. We may now agree to represent the line drawn from the 

 origin to the point whose rectangular co-ordinates are x and 

 y, by the binomial x + s[y). The first consequence of this will 

 be, that the sum of two lines will be represented by the dia- 

 gonal of the parallelogram whose sides are the lines to be 

 added. In fact, analogy so strongly demands this, that it is a 



