Determinantal Equation, 221 



has all its roots real, and thereupon adds that u a somewhat 

 similar process shows that the roots of the equation 



a — x 

 d 



9 



b 



e—x 

 h 



c 



f 



i — x 



= 



are always all real, provided the single condition 



cdh — bfg 

 be satisfied." 



On examination of this latter statement in the light of 

 former researches of my own, I found that it was scarcely 

 correct to say that only one condition was necessary, the 

 further requirements being that b and d be of like sign, c and 

 g of like sign, and therefore also / and h of like sign, — in 

 other words, that the determinant should, so far as sign is 

 concerned, be axi- symmetric ; and from this I passed to the 

 consideration of similar equations of higher degree, with the 

 following results : — 



2. Taking the equation of the rcth degree, but for short- 

 ness' sake writing it only of the fifth, viz. 



a ± —x a 2 



bi b 2 — x 



C\ c 2 



di d 2 



e x e 2 



a 3 



h 

 c 3 — x 



d 3 



a 4 



a 5 



h 



h 



Ci 



C 5 



d±—x 



d 5 



<?4 



e 5 — x 



= 0, 



and multiplying the rows by © l3 co 2 , © 3 , co 4 , a> 5 respectively, 



and the columns by co x l , © 2 1 J ©~ 

 have the equation in the form 



i 



4 J 



© 5 respectively, we 



a x — x 



a\co- 1 b 1 



©4©-^ 



©i© 2 l a 2 

 b 2 — x 



©3©~ 1 C 2 

 ©4©~ 1 d 2 

 ©5©-^ 2 



(D l a)- l a z 



©2©^ 1 &3 



c 3 —x 



© 4 ©-W 3 



to 5 (o~ l e 3 



©!©" 



©x© 5 J a 5 



<*>&>4 l h 



© 2 ©- l b b 



©s©^ 1 ^ 



G>3«r^5 



d±—x 



© 4 © 5 -^ 5 



©s©- 1 ^ 



e b —x 



= 0. 



The determinant here is in substance exactly the same as 

 before ; but we have now Hve disposable quantities, ©j, © 2 , 

 © 3 , © 4 , © 5 , and the question is whether these can be so deter- 

 mined as to make the array of elements axi- symmetric. The 

 conditions for this clearly are 



> x © 2 l a 2 = <i) 2 b} '- 



©i© V/ 3 =© 3 © V b 



