so Proceedings of the Royal Irish Academy. 



In Art. 6, O was expressed as a sum of 2m - 1, or 2m binary pro- 

 ducts ; it is now reduced to a sum of m binary products. This reduc- 

 tion leads up to tlie investigation contained in the following article. 



9. The general problem announced in the last article may be 

 enunciated thus: — 



Given any homogeneous function of any number {N) of the n 

 vector units consisting of a sum of products of any number [m) of 

 distinct units, each multiplied by a given scalar, to reduce this func- 

 tion to a canonical form by a change of the system of units involved. 



Let q be the given function (of order m), and «i one of the units 

 involved. It may be written in the form q = - q'ii + q'\ in which q' 

 and q" are both independent of ii. ilultiply into ?i, and 



qh = '^mn^ii + ^m-ij^'i = 2' + q"ii 

 ^ives separately 



q' = K^iqiu and q"ir = Kn^iqh- 



IS'ow multiply q into q', and take the part ( T^iqq') of the product 

 ■qq\ which is linear in the units. This new vector (■uJj) will not, in 

 general, be parallel to ii ; but it is a linear function of ?'i, expressed by 

 •the formula 



The linear vector function $ defined by the equation 



^p = Viq ^m-iqp 



is easily seen to be self- conjugate, for 



Sa-^p = S<jViqV,,,_^qp = S(rqV,„_^qp = SV„,-^<Tq . V,„_iqp 

 = >S' V,n.iqcr V„,^ipq = Sq V,n_^qa- .p = S V^q V,n-iqcr . p = S^a . p. 



4> being self-conjugate, just as in quaternions, its axes are all real 

 and mutually rectangular. These axes are the units to be employed 

 in the reduction to the canonical form. 



10. As an example, consider the reduction of the general homo- 

 geneous quadratic function of JV of the n units. It consists of 

 ■JiV(iV- 1) binary products, each of which is multiplied by a 

 scalar. 



Then q = zj^i^ + q', suppose, where neither w^ nor q' involve ii. 

 JSere 



- -57^ = Viqi, and Fj^^i = - /i-3?i^ + Viq'zj^ = - Viq f-^qii = - $«i. 



