384 Proceedings of the Royal Irish Academy. 



and, because- . 



. = 0, or a{Af\ + ^{B/)„ + y.{Cf)„ = 0, . . 



{Af\ = {Bf)„ = {Cf\ = Q. 



If a, /?, and y are taken to be the principal axes of the quadric 



ISp(r'p = - 1, 



the additional simplicity of the mutual rectangularity of the coordi- 

 nating vectors is obtained. 



10. Introduction of a second invariant, ivkich, cannot generally he made 

 to vanish when n is odd, and is then a vector. 



Again, consider the invariant (12)", obtained by operating "with the 

 operator derived from the quantic 



{^^,(3,...l3„){xyr 



on the quantic itself. This is, by the principles of Art. 4, 



ySoiSn - ^^Ai3„-i + . . . + (-)"/3„i8o. 



First, taking the case in which ?i is odd, the invariant is a vector, 

 and its half is 



V{/3J3„ - n{3,(3„., + &c.), 



For example, if the binary quantic is 



p (ffloOi . • . an){xyf - (ttotti . . . a„){xyf, 

 the invariant is 



^[(p«o - ao)(p«n - On) - » (pfli - ai)(pff„_i - a„_i) + &C.] 

 = Vp [a^a„ - a„a^ - n {a^a^.^ - a„.iai) + ...]+ r(a„a„ - ?mia„_i + &c.) 



= Fpt + K 



in the notation of Art. 7. This invaiiant cannot in general be made 

 to vanish by change of origin of vectors ; if it vanishes, Slk = 0, and 

 this is not generally true. In fact, it is easy to see that, on change 

 of origin, the invariant k becomes k + Vp^L, where p^ is the vector to 

 the new origin, and thus the scalar Skl is quite independent of the 

 position of the origin, as it has been shown already that t does not 

 change with change of origin. 



