68 



Proceedings of the Royal Irish Academy. 



III. — Eeduction to Canonical Foem. 



18. A proof based on equality of numbers of independent co-ordinates 

 (such as in § 16) is not always reliable, though it produces a high degree of 

 conviction. It is necessary, therefore, to show how the reduction to the 

 canonical form may be effected directly, and further that it is unique, 

 provided there are no singularities.* 



Let / (X) represent the determinant 



/, -X 



mi 





«i 



Pi 



mg - 



X 



ih 



Ih 



m. 





«3- X 



Ih 



TOi 





n. 



i'l 



The coefficients of X in this equation of the 7i"' degree are invariants for 

 all orthogonal axes. For, let F {x, y, z, u, v, . . .) be a point which becomes 

 transformed to P' (Kx, \y, \z, Xu, \v, . . .) ; theuf 



Xx = l^x + mil) + etc., 



Xy = kx + nhy + etc., 



etc. 



If we transform to other orthogonal axes through the point, it is clear 

 that if X, Y, Z, U, . . . are the new co-ordinates of P, then XX, X Y, XZ, . . . 

 are the new co-ordinates of P'. Hence the equation /(X) = formed 

 from the new coefiicients must have the same roots ; and since the absolute 

 term in f{X) is the same in both cases (A = 1), all the coefficients of X are 

 absolute invariants. Now, if we take the canonical form of § 17, it is easily 

 seen that /(X) vanishes when X = e-'^ . For the left-hand top small minor 

 becomes 



icos ^ - X sin B I 



= (X-e«)(X-e-'«), 

 - sin 6/ cos y - X | 



and if this vanishes /(X) = 0, since the determinant is simply the product 

 of these small diagonal minors. 



Thus, wAew w is even, the roots of f(X) = are the n quantities e*'*, e*'*, etc., 



whereby the -z angles of rotation are determined. When n is odd, the corre- 

 sponding roots of f(X) = a.re n - 1 quantities e-'^ and unity. + 



* See Note, p. 73, and footnote p. 70. 



t Constants a, b, c, . . . may be added, but it makes no difference. 



X Because when n is odd /(I) = always (see § 18). 



