466 Mr. J. J. Sylvester on Inverse Orthogonalism 



Phases of nucleus to type 2.2: — 



1 1 + 

 + 1 1 

 1 + 1 



+ 



- - + 



- + - 



+ 



- + - 

 + 



+ 



— "•+■— 



+ 



+ 



+ 



+ 



+ - - 



- + - 



Phases of nucleus to type 4 : 











i — i' 



- + - 

 i' — i 



— i' i 



+ 



- i i' 



H i — 



- - + 



i a — 





i — i' 

 V — i 



+ 



— i 1 i 



— i i' 



- - + 

 i ! i — 

 i i' — 



— i i' 

 + - - 



— i 1 i 



i' — ( 

 - + • 



i — i 1 



i i' — 

 i' i — 



+ - - 



— i i' 



— i' i 



- + i 

 i 1 — i 

 i — i 1 



i i' — 



i 1 i — 



i — i' 

 i' — i 



_ + _ 



— i' i 



— i i' 

 + 



i' i — 

 i i' — 

 + 



— i i 1 



— i 1 i 



+ 



i 1 — i 

 i — i f 



- + - 



i i' — 

 i' i — 



Thus, then, the total number of distinct solutions of our 

 (4 — l) 2 , i. e. 9, algebraical equations applicable to this case is 

 18 + 6, or 24. The formula II (ft— 1 ) .TL(n— 2) would give only 

 12. How it should happen that the order of the system of 

 equations for different values of n is not an algebraical, but a 

 transcendental function of n depending on the factors of which 

 n is made up, will become less surprising when it is considered 

 that the quantities equated to zero in any such system, although 

 algebraical in themselves, are not analytical but tactical functions 

 of n their degree. 



7. It remains to assign the value of the constant product in the 

 reduced form of matrix of the order n, or, which comes to the 

 same thing, the value of the complete determinant of such ma- 

 trix, which is obviously n times the former quantity. 



(1) When n is undecomposed, the value of this determinant, 



