84 Proceedings of the Royal Irish Academy. 



The quadratic may be reduced to amij^ + au^^is, and therefore 

 the cubic may be reduced to ai2-iiihi3 + cin^iiiiis. 



Generally, by this process, an m^" in m + 1 units may be reduced to 

 a linear vector multiplied by o>, and the wi'" may consequently be 

 replaced by a single product of m units multiplied by a constant. 



Also, an m^" in m + 2 units may be reduced to a quadratic in the 

 same number of units multiplied into w. It has already been shown 

 how to reduce a quadratic, so the w'^ in m + 2 units may be considered 

 known in the canonical form. 



14. This process does not apply to a cubic in six units, for a cubie 

 in six units is reproduced. 



It is easy, by the aid of a geometrical method, to -write down 

 examples of cubics reduced to the standard form. Take any three 

 points, 1, 2, and 3; they may be joined to form a triangle (123). 

 Take a fourth point, 4 ; every triangle formed with this point and a 

 pair of the old points has a side common with the triangle (123). In 

 the last article it was shown that the cubic in four units is reducible 



to ^123^1^2^3 • 



Pursuing this analogy, consider how in a few simple cases a limited 

 number of points can be joined to form triangles which have no side 

 common to two. Adding a fifth point 5 to the four points 1, 2, 3, and 4, 

 but two triangles, (123) and (145), having no side common, can be 

 drawn connecting these five points. (Of course, other pairs of tri- 

 angles, (512) and (534) for instance, may be drawn to connect the 

 five points. This is merely a matter of selection.) In the last article 

 it was shown that a cubic in five units is reducible to «i23h44 + (lubiiHio- 



Adding a sixth point to the five already taken, and two different 

 arrangements are possible. Either four triangles (123), (145), (624), 

 (635), or a pair of triangles (123), (456), can be drawn connecting: 

 the six points, and having no side common. 



It may be verified at once that the cubic 



q, = ctmiiiiiz + (iii-JiiJo + au^iziih + (tstshioh 



which corresponds to the first of these arrangements is in the canonical 

 form, and so is the cubic 



q' = a^^zhizh + aiseUhh- 



These cubics belong to distinct types, and cannot be transformed into 

 one another. The square of q' is a scalar {a^^z + «\5g), but the square 

 of q is not a scalar. 



