92 Proceedings of the Royal Irish Academy. 



In general, the conditions of Art. 20 holding true, 



qKq = q^^f - q^^^f - q^^f + q^^f + £(0)^(3) + <?(3)^(o) - ?(i)$'(2) - ^(2)^(1) 



= ^CO) (^(0)' - £(.)' - ?(2)' + ^(3)') + 2 r(3) (^(0)^(3) - <?(1)!?(2)), 



the insertion of the symbols ^(^3 and V^z) heing Justified by the con- 

 sideration that the function is its own conjugate. If it reduces to a 

 scalar, the odd part must vanish, or 



2'(0}2'(3) + 2'(3)2'(0) = 2'(1)2''2) + 2'(2)2'(l)5 



and also ( V^,^ - V,) {q^^f - q^^f - q^^f + q^zf) = 0. 



23. Considering specially in the first case a quadratic in the units, 



qKq = {q, + q, + q,) {q, - q, - q,) = q,~ - {q, + q.f = Kq . q. 



If this i)roduct reduces to a scalar, the part which is odd in the units 

 must vanish, or q^q-i + q^q^ = 0. 



Let g'l = al^l, and q-i = ciyiixii + )8, 



where ^ does not involve ix ; then, if a^ is not zero, 



ii^ + y8^■l = 1i^^ = 0, 

 so /3 must be zero, and the function is reduced to the Quaternion type 



q = aQ + (hii 4- ai3*>2, 



and involves but two units. It is evident that qz' is for this a scalar. 

 Again, if qi vanishes, suppose g'2 reduced to the canonical form 



(I'i2iih + ttnhh + &C. 

 Squaring, it is found that 



q^ = - ai-^ - «3/ - &c. + 2ai2a$iiii2hii + &c. 



And this will not reduce to a scalar, unless all but one of the coefficients 

 «i2 vanish. So again, g-j = ^g + «i2«i4 is of the Quaternion type. 



The theorem is thus proved that, if the product by its conjugate ef 

 a quadratic function in any number of units is a scalar, the function is 

 capable of being reduced to the Quaternion type involving but two units. 



24. In the second case for a cubic function, in addition to the 

 equations of commutation which reduce to 



SiSs - q^i = 0, and q^q^ - $-3^2 = 0, 



because q^ is a scalar, the conditions that qKq = Kq . q should be a 

 scalar, become 



2M3 = m^ + M^ ^'^^ ^^ - li = scalar. 



