and Simultaneous Sign-succeseions. 463 



4. This result is easily verified. For, reverting to the example 

 of the third order, if any inverse orthogonal matrix of that order 

 is multiplied, term to term, by the following one, 



Tk } lfjb } iv, 



m~k, mfju, mv, 



rik, n/jt,, nv, 



the product so formed will evidently retain its character unal- 

 tered, since each of the equal products will receive a constant 

 multiplier, Imn . Xfiv. 



The number of independent quantities thus introduced is 5, 

 viz. 



a; TV \'x' 



and so in the general case we can introduce (2n—l) arbitrary 

 elements. Thus, then, we may without any loss of generality 

 regard only those matrices of the kind in question which are 

 bordered horizontally and vertically by a line of positive units. 

 From these reduced forms it is easy to pass to the general forms 

 by term-to-term multiplication with a matrix of the kind above 

 denoted. The question now becomes narrowed to that of deter- 

 mining the number and form of the reduced inverse orthogonal 

 matrices of any given orders, — a problem (if attacked by a direct 

 method) involving the solution of (ra— l) 2 equations between 

 (n — l) 2 unknown quantities. 



5. (1) Let n be a prime number. Write down the line of terms 



1, «, 



and make a equal in succession to each of the (ft — 1) roots of 



£n \ 



— =0. The matrix so formed will be a reduced inverse or- 



# — 1 



thogonal matrix of the ?zth order. 



In the case of n = 3, it is easy and will be instructive to verify 

 this statement. Calling the required matrix 



111 

 1 a b 

 led, 

 we obtain the four equations 



ad-bc=d(a-\)=c(l-b) = a{d-\) = b{l-c) } 

 which are equivalent to the following, 



ad=c = b, bc = d=a. 

 Hence 



a?d 2 = bc, or a 4 = a. 

 Hence rejecting the values « = and « = 1, either of which 



