94 Proceedings of the Royal Irish Academy. 



For a quadratic, the conditions reduce to q^q^ - qiq^ = 0, or §'2 must he 

 indej)endent of the vector qi. If qi vanislies, the condition i-s identically 

 satisfied, 



Por a cubic, q.. must be independent of qi ; qi must be a factor of ^3, 

 if it does not yanish. Otherwise, qo and q^ may be arbitrarily chosen. 



26. If qlq = Iq . q is a scalar, it is necessary that 



S,o)^ + 2w - 5'(2)^ - S'o/ = scalar, 



and that ^(o)^(i) + ^(d^co) = ^(2)^(3) + S'(3)?(2), 



or that r(o) {q,J + q^^f - q^^f - q^^-) = scalar 



and Ffj) {qiomD " S^Si^) = 0. 



For a quadratic, qi or ^'g must be zero, and q^ must be az^izh, as its 

 square is scalar. The types are 



q = (hh + aoJzh, or q = aQ + a-oJoH. 

 For a cubic, 



q^qi = ?i?3, ?2^i = qiqz, 2^1 = ^2^3 + ^3^2, 



and qz + qi = scalar are necessary. 



If qi = «i?'i ^3 = iifS, and $'2 = /3' does not involve ?'i. Also, 



2ffoffi = ySyS' + /3'/3, and ^'- - /3^ = 5c«?^r. 

 Reduce fS to the canonical form 



/3 = 323/2«3 + hohh + • . ., 



and from the first condition it is seen that, in order to be rid of iihhhj 

 &e., it is necessary that /3 = ho^ioji + bi^ij^, and 



13' = S (523'44 - ^45«V4) + J'2i4?i + ^'25?24 + i'sihii + V Tohh- 



The second condition requires (for real functions) §45 = 0, and jB' may 

 be reduced to /3' = I'izhiz + V'iJzii + VziizH- 



Thus, ^3 = amhioJz, and $'2 = ffjs^^s + az^hh + fl!42^'i4- 



27. If any product (^;) of linear vectors is formed, it is obvious 

 that the conditions 



2)JSjp = K2) .p - scalar., and jyT^ = Ip .p = scalar 



are both satisfied. It would be desii'able to prove or to disprove the 

 statement that any function p satisfying these conditions must be a 

 product of linear vectors.^ 



1 See p. 96. 



