468 Mr. J. J. Sylvester on Inverse Orthogonalism 



we may represent the values of A and B by writing the three 

 matrices 



« /3 





a b 





+ + 



« 





b a 





+ — 



on multiplying these three together, term by term, we obtain 



+ aa-\-/3b 



+ ub-\-/3a 

 where 



■\-ua + @b = k, 



-f ub— /3a = B. 



Moreover the term-to-term product of the second and third 

 a. b 



matrix, viz. 



b, —a 



So again in the equation 



(a 2 -f/3 2 + 7 2 + S 2 )(a 2 + Z/ 2 + c 2 + 

 the three matrices become 



, is an orthogonal matrix. 



= A 2 + B 2 + C 2 + D 2 



a ft 7 8 





abed 





a ft 7 8 





bade 





a ft 7 8 





c d a b 





a ft 7 8 





d c b a 





+ 



+ 



+ 



+ 



+ 



— 



+ 



— 



+ 



— 



— 



+ 



+ 



+ 



— 



— 



The resulting product, 



cta-hftb-\-<yc + 8d 

 xb — fta-\-<yd—8c 

 ac—ftd—<ya + 8b 

 ctd-\-j3c—<yb — 8a } 



represents in its four lines the respective values of A, B, C, D. 

 Moreover the matrix produced by the product of the second and 

 third, i. e, 



a, b, c, d 



b, —a, d, —c 

 c } — d, — a, b 

 d, c, —b, —a 



is an orthogonal matrix. The same remarks apply to the re- 

 presentation of the product of two sums of eight squares under 

 the form of a sum of eight. Omitting the first matrix, consisting 

 of repetitions of one given set of eight letters, a, /3, 7, 8, X, //,, v, tt, 

 the remaining two matrices may be written as below : 



