the others bein£ 



Deter minantal Equation^ 22S 



a 2 — b ] 



«3 • ~ <?1 =0. 



a 2 h z c 1 =a 3 b 1 c 2i I 



^2^46?! = aj} x d 2 , 

 a 2 5 5 ^i=a5& 1 ^2j 



«3C 5 ^ =a 5 C!e 3 , 

 a±d 5 e 1 =a 5 d 1 e /L , 



Further, no other conditions are necessary : for, these being 

 complied with, the values of the ratios obtained from the first 

 four equations will satisfy the remaining six. 



3. Any conditions of consistency, therefore, which may 

 be obtained from the last six equations only must be depen- 

 dent on the conditions already obtained. For example, from 

 (5), (6), (8) we have the condition 



b 3 c^d 2 = b^c 2 d 3 ; 



but this is merely a result derivable by multiplication and 

 division from three of the previous six conditions, viz. the 

 three 



a 3 c 4: d 1 = a 4 c 1 d 3 , a 2 b 3 Ci = a 3 b 1 c 2) a 4 b 1 d 2 = a 2 b 4 d 1 . 



The number of such dependent conditions is four, viz. 



from (5), (6), (8) hc A d 2 = b^d 3i 



from (5), (7), (9) b 3 c 5 e 2 =b 6 c 2 e 3 , 



from (fi), (7), (10) b i d 5 e 2 =b 5 d 2 e 4 , 



and from (8), (9), (10) e^d h e 3 = c b d 3 e±. 



4. Again, the ten conditions regarding the signs of the 

 elements, viz. 



«2^1=+> <%=+, 



are not all independent. Only the first four are necessary 

 when taken along with the six equations of § 2. For the 

 conditions 



aj) z c x = a 3 b x c 2 , 



a 2 bi — + , 



a 3 c 1 = + 



give b 3 c 2 =+, 



a 2 b^d Y ~a^) X d^ 



a 2 b { = +, 



«4^1= + 



give M 2 =+, 



a 2 b 5 e 1 =a 5 b 1 e 2 , 



a 2 b x = +, 



«5«1= + 



give V' 3 =+, 



b 3 c 4 d 2 = b±c 2 d 3 , 



<V2=+, 



Ms = + 



give ^3=+, 



~b 3 c h e 2 =b h c 2 e 3 , 



b 3 c 2 = 4- , 



^2=+. 



give c b e 3 =+, 



l\d b e 2 =b b d 2 e±, 



hd 2 = + , 



^2= + 



give d b e±=+. 



