90 Proceedings of the Royal Irish Academy. 



In the case of a cubic function of the units, it is necessary that 



f^{o)M^ = 0, and r(3)^22'3 = 0, 



or more simply that 



^4$'i2'3 = 0, and Vsqoqs = 0. 



If the cubic is q = do -^ 2«i«i + ^aioJiiz + ^ai^ziihh, 



these are Viq^q^ = % {aiO^u - ((^(fm + dzC'm - dif^m) iihhh = 0> 



and Vsq^qs = - 2^ {dsia^^s + a^a^^i + a^a,^^) i^i^H = 0, 



and they are identically satisfied for a cubic in three units, that is, for 

 the general function of three units. 



Generally for a cubic, let ii be defined by the equation q^ = aiii ; 

 then, provided «i is not zero, i^q^ - q^ii = 0, that is, ii is commutative 

 with a homogeneous cubic in the units. Consequently, this cubic must 

 contain i^ as a factor, or q^ = iiq'i^ where q'^ does not involve ii. Turning 

 to the second condition, supjDose q^, = (hzhit + 9,"z^ where q'\ does not 

 involve e'l, and 



= «i2 {Hq'2 + q'ih) + ii {fiq'-i - q'2q"-z) = 0. 



This requires separately ^^''a + q'2t2 = 0, or 4 must be a factor of q'2, or 

 q'i = duzhh) say, provided «i2 is not zero ; and also q"2q'2 - q'zq"^ = 0- 

 This last reduces to (^'2i%i% - i^izq''^ = 0, and making the legitimate 

 assumption 



q"2 = (lisiojs + (('■zii-Ji + q"'2 ; 



where q"'2 does not involve 4, it further reduces to 



- ^23 + (fzJJz + q"'ihi3 + C'23 - a2ihii " hhq"'2 = 0- 



This requires azi = and q'"2 to be independent of is, and the cubic is 

 reduced to 



q = ao + a it I + «l2^l4 + ^23^-3 + «i23h«24 + q"'2, 



in which q'"^ is independent of «i, «2, and ?3. 



If, however, «io is zero, the condition is q2q'2 - <^2q2 - 0> in which 

 both functions are independent of i^. Let (^2 be reduced to the canonical 

 form, so that (^2 = ttmi2i3 + ^'"'2? iii which q'\ is independent of i^ and 4, 

 and let 



as before. Then, as formerly, «24 is zero, and $'"'2^3 = 4?"'2) or q'"^ is- 

 independent of ij. 



