244 Mr. DE morgan, ON TRIPLE ALGEBRA. 



§ 3. Simple and perfect cubic form. I now proceed to consider the simple cubic form in 

 case B. The equations of signification* are (dropping the distinctive symbol ^, which is in- 

 operative), 



And the product of a + bt) + c^ and a' + b'tj + c'^ is 



be + cb' + aa + (ab' + ba -cc) rj + (an + ca - bb') ^. 



If the equations of signification be also consistently algebraical, and if >; = ^ and ^ = >" satisfy 

 them, then a +bij. + cv is a modulus of multiplication. Accordingly in the present instance, it 

 is sufficient that fx and v should be severally equal to — 1, or else that they should be the 

 imaginary cube roots of - 1. Let them be the latter : then a — b — c, a + /mb + vc, a + vb + /xc, 

 are moduli, and since any product of roots of moduli is a modulus, we have, taking such roots 

 as are required by the condition that the Algebra is to become single if b and c always vanish, 

 the following possible moduli, 



a — b — c, 



y/ {a' + b^ + (f + ab + a c - be), 



x/{a^- h'-c'-Sabe). 



These expressions are connected with the third degree in the same manner as a^+ 6- with the second. 

 Changing the signs of 6 and c &c., their modular character gives the following equations. Let 



A = be + cb' + aa\ B = ab'+ba + cc, C = ac'+ ca'+ bb'. 



Then (a + 6 + o-) (a'+ b' + c') = A + B + C 



{a^+b'+c^- ab -bc-ca) (a'' + b'^+c'^'-db' - b'c'-c'a) = A'+ B- + - AB - BC - CA 



{a?+W + c^-3abc) {d^ + 6'' +c'^-S a'b'c) = A^ + B" + C - 3 ABC. 



These might, I think, be made of the same sort of use in the theory of numbers with the equation 

 (a'+ 6') (f''"'+ 6'") = {ad - bb'y+ {ah' + ba')-, which is the modular equation of the common Algebra. 

 Thus of either of the forms d'+ 6^+ c'— ab - be - ca and a'+ 6^+ r*- Sabe we may say that the 

 product of two instances must be a third instance. 



It appears that this cubic form of triple algebra may involve three cases, according to 

 the modulus which we employ. Now we know that in common Algebra, a + b^y - 1 is made to 

 depend upon a length and an angle, in such a manner that the length is represented by the modulus, 

 and the product of two expi'essions has the product of the lengths for a length, and the sum of 

 the angles for an angle. Suppose that we make a + ?)>; + c^ to depend upon the modulus and 

 two angles, each having the same property as the angle of the former case : it is required to 

 express a + 6»j + c^ by [I, 6, (p\ in such manner that the following equation may be identically true, 



[/, e, <p] . [/', &, <p'] = [W, e + e',^ + cp']. 



Without as yet specifying which modulus we are to take, we must examine into the conditions 

 of a species of triple trigonometry, in which two angles form the base of every expression. 

 Looking at the form of the product of a + br/ + c^ and a'+ 6't; + e'^, it is obvious that the 

 problem is solved if we can assign 



A -- B -'' C -- 



• In this sense it ought to be remembered that they more resemble — x — = + than ab = c. 



