224 Dr. T. Muir on Lagrange s 



5. The state of matters thus is that the quintic equation 

 a x —x a 2 ... 



bi b 2 —x 



= 



will have all its roots real if ten conditions be complied with: 

 viz. 



six as to magnitude, and four as to sign. 



a 2 b 3 c i = a 3 b 1 c 2 , 

 a 2 b±di = aj)±d 2 , 

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

 a 3 c i d 1 = a^c 1 d 3 , 

 a 3 c 5 e 1 =a 5 c 1 e 3 j 

 a 4 d- 5 e 1 = a 5 d 1 e 4: , 





and that the six conditions as to magnitude imply four others 

 like themselves, viz. 



b 3 C4d 2 = b A c 2 d 3 , 



b 3 c 5 e 2 = b 5 c 2 e 3 , 



hd 5 e 2 =b 5 d 2 e^, 



^d h e 3 — c h d 3 e^ 



and the said six together with the four conditions as to sign 

 imply six others similar to the latter, viz. 



hc 2 =+, 

 b A d 2 — + } 

 b 5 e 2 = +, 

 c A d 3 = +, 



d 5 e^ = + . 



6. The law of formation of the magnitude conditions is not 

 readily apparent, hut a change to the double-index notation 

 for the elements of the determinant suffices to clear up the 

 difficulty. 



Writing | a^) 2 c 3 d^e h | in the form 



11 



12 



13 ... 



21 



22 



23 ... 



31 



32 



33 . . . . 



