JoLY — The Associative Algebra applicable to Hyperspace. 91 



Hence, it is proved incidentally that, if two homogeneoug quadratics 

 are commntatiye in order of multiplication, they must he reducible 

 simultaneously to the canonical form. 



Thiis, when ai^ is zero, the cubic commutative with its conjugate 

 must be of the type 



q = (10 + (tiii + q2 + ^i^'a, 



in which ^2 ^^^ s'z are simultaneously reducible to the canonical form. 



"Next, if «i is zero, or if the cubic is q = qo + q2 + 2'3> the condition 

 becomes q^q^ = q-^qo ; and it is necessary to ascertain under what circum- 

 stances a homogeneous cubic and quadratic can be commutative. 



In the first place, they are commutative, if they have no unit vectors 

 common. Again, a cubic linear in the units common to the quadratic 

 cannot be commutative with it. Thus, the cubic may be reduced to 

 the form q^ = '^Jifii + y + y'j where y' is a function of the units/ which 

 do not occur in the quadratic, where the quadratics (3 are functions of 

 the units occurring in the quadratic and simultaneously reducible with 

 it to the canonical f o]Tn (or the /? are commutative with the quadratic), 

 and where y is a function of the units in the quadratic alone. 



It is only necessary to consider the case in which the function q^ 

 involves no units not contained in the quadratic q^. 



iN'ow, it is easy to show, when the quadratic is reduced to the 

 canonical form, 



2'2 = (tniiii + «3i«3«'4 + &c., 



that the cubic can contain no term involving a product of conjugate 

 units (^"i and 4, or 4 and ii). For, suppose the vectors ii and 4 to be 

 explicitly expressed in the equations of the cubic and quadratic by 



S3 = o.iziiiz + /?i?'i + /S2/2 + y? snd q^ = aniiiz + yS', 



^2^3 - M2 = (/S'«i2 - «i2/3') iiiz + (/3'ySi - y8,/3' - 2«,2^o) i, 



+ (/3'A - /?2^' + 2a,2^0 H + ^'y - yiS' = ; 



and this requires /3'ai2 - ai^f^' = 0, which cannot hold if the vector ai2 

 is different from zero, for the case in which a^^ is independent of the 

 units in /3' has been specially excluded. 



I see no simple step towards completion of the problem. 



22. Eeturning to the value of 



qKq = Kq.q = q'^ - q"^ = {q^,, + q^,.)'^ - {q^.^ + ^'(2))^- ; 



when the conditions of Art. 20 are satisfied, it is important to inves- 

 tigate the conditions that this product should be a scalar. Tor, if q is 

 a product of functions such as c/q + 2«i/i, qKq is necessarily a scalar. 



