462 Mr. J. J. Sylvester on Inverse Or 



racter of orthogonalism in the matrix of substitution employed. 

 Consequently the number of conditions to be satisfied is the 

 number of terms in a homogeneous quadratic function of n 



variables, which is ' 9 -. In an orthogonal matrix (not 



strictly so) the number of implications is consequently 

 (n + 2)(n-l) 



3. The problem of constructing an inverse orthogonal matrix 

 of any order admits of a general and complete solution. It is 

 to be understood in what follows, that the constant product of 

 any term by its first minor is not to be zero ; or, in other terms, 

 the complete determinant of the matrix which is a sum of such 

 products is not to vanish. 



First, let us investigate the number of arbitrary elements which 

 enter into any such matrix. 



To fix the ideas, consider one of the third order, say 



a b c 

 * fi 7 

 ABC 

 and call the reciprocal matrix formed by its first minors 



Then 



a 



l u i 

 Ax B x 



7i 



aa. 



= bb } 



cc-, 



= otoi l =#& =yy 1 

 =AA 1 =BB 1 =CC 1 . 



These 8 equations are not independent ; for we have 

 aa Y -{•bb l -\-cc l =.aa l -\-uc*. l -j-AAj 

 = ««! +/3/9J + 771 ^bb 1 + ^/3 l + BB l 

 = AA 1 +BB 1 + CC i = c£ 1 H-77 ] + CC 1 ; 



which 5 equations in their turn again are not independent, be- 

 cause the sum of the three groups written under one another on 

 the left is equal to the corresponding sum on the right. 



Hence we have implication upon implication, so that the num- 

 ber of independent equations is 



(32_ 1 )_(2.3-1) + 1 = (3-1) 2 ; 



and so in general for a matrix of the order n, the number of in- 

 dependent equations is (n — l) 2 , leaving 2?z — 1 of the elements 

 arbitrary. 



