and Simultaneous Sign -successions. 467 



by virtue of a well-known theorem given years ago by Professor 

 Cayley, for expressing the discriminant of an algebraical function 

 as a determinant composed of powers of its roots, is easily recog- 



n 



nized to be i(»-i)(»-2)^ which we mav call A n . 



(2) When n is decomposed under the form p, q, t,.,., the 

 corresponding determinant may easily be proved equal to 



Hence the determinant in the general case is 



qpr... rpq 



where 



( — )*p -* .q 2 .r 2 ... =(-)Qn 2 , 

 P 



Thus, if each term in any reduced inverse orthogonal matrix 

 of the order n be divided by the square root of n } the fourth 

 power of the resulting determinant is unity for all the types with- 

 out distinction. If n is decomposed into p, equal factors p } 



<l>=f*(p-l)(p-2)p»- 1 ', 



so that when p, > 1, the determinant is + i if p, = 1 [mod 2] , and 

 pz=. — 1 [mod 4], and is +1 in all other cases. When /^=1, 

 its value is (- + i) if p=L — -1 or [mod*4], and +1 in the other 

 two cases. When n is undecomposed,the value of the constant pro- 

 duct, which is - of the determinant, takes the simple form 

 ' n 



n-2 



(i n - l n) 2 . 



8. When n is a power of 2, the type corresponding to its de- 

 composition into the equal factors 2 deserves especial conside- 

 ration. In this type the only roots of unity which appear are 

 1 and 1; and as each of those numbers is its own arithmetical 

 inverse, the matrix may be said with equal propriety to be in- 

 versely orthogonal or directly orthogonal, i„ e. orthogonal in the 

 sense conveyed in art. 1. Moreover, on dividing each term by 

 Vn, it becomes strictly orthogonal, since the sum of the squares 

 of the terms in each row or column then becomes unity. 



A very little reflection will make it clear, a priori, that using 

 simply + and — in place of + 1 and —1, the known theorems 

 relating to the form of the products of two sums of 2, or of 4, 

 or of 8 squares must exhibit instances of orthogonal matrices of 

 this nature. Thus, to begin with the case of the equation 



