42 = 32.43.24, 

 52 = 32.53.25, 



52 = 42.54.25, 



53 = 43.54.35. 



Delerminantal Equation. 225 



the conditions in question are 



12.23.31 = 21.32.13, 23.31 



12.24.41 = 21.42.14, 23.35 



12.25.51 = 21.52.15, 24.45 



13.34.41 = 31.48.14, 34.45 

 13.35.51 = 31.53.15, 

 14.45.51 = 41.54.15, 



Each condition is thus seen to have its origin in a triad of 

 the five indices 1, 2, 3, 4, 5. For example, from the triad 123 

 we form the elements 12, 23, 31, and from these the conju- 

 gate elements 21, 32, 13, and so arrive at the condition 



12.23.31 = 21.32.13, 

 which, in order that the distinctive character of its formation 

 may be more apparent, would be still better written 



1 2 3 __ 2 3 1 

 2'3'l ~ l'2'3" 



In this way the number of conditions is seen to be C 5>3 , 

 the necessary conditions being C 4j2 in number, and the depen- 

 dent conditions 4j 3. 



7. The general theorem may consequently be enunciated 

 as follows : — 



The nthic equation 



11-a 12 13 



21 22-^ 23 



31 32 33-0 



= 



will have all its roots real, if in the case of every pair /-t, v of 

 the indices 2, 3, 4, . . . , n we have 



and 



1 A 6 v _ A 6 v 1 



/*' V* 1 ~~ 1 * fjb'v' 



V> - 



= + 



these conditions implying that in the case of every triad fj,, v, p 

 of the indices 1, 2, 3, . . . , n we shall have 



and 



A 6 v 9 _ P A 6 v 

 p ' fi'v ~ fi' v'p' 



** v = + 

 v'fju 



