222 Da\ T. Muir on Lagrange's 



or, better, 



w^a 2 = (D 2 2 b 1 , w 1 2 a 3 = a) 3 2 c I , aj 1 2 a 4 = oj 4 2 rf 1 , co 2 a 5 = (o^e 1 : 



(0 2 % = (0 B C 2 , a) 2 2 6 4 =&> 4 2 d 2 , <02% = a>b e 2\ 



co s 2 c± = a) 4 2 cZ 3 , co d 2 c 5 = co 6 2 e B ; 



G) 4 2 dg = a> 5 2 <? 4 : 



from which it is evident that, whatever the full and final 

 answer to our question may be, it will be necessary that 



a 2 have the same sign as 6 1? 



a 3 D J? JJ G ll 



— in other words, that conjugate elements of the original deter- 

 minant be alike in sign. 



As, however, we have got to ascertain whether a set of non- 

 zero values can be found for co l , co 2 , g> 3 , o) 4 , co 5 which will 

 satisfy all the equations, it is desirable to arrange them for- 

 mally as a set of (ten) homogeneous equations in these iive 

 unknowns. The result of this is 



a 2 co L 2 - 



- W(0 2 2 









= 0, 



«3«1 2 





~C x <Oi 







= 0, 



<V>1 2 







— dyW^ 





= 0, 



a 5 co{ 2 









~ ei<» 2 



= 0, 





h«»-2 2 



— c 2 (o 3 2 







=0, 





b 4 w 2 2 





— d 2 <0± 2 





= 0, 





b 5 (o 2 2 







-e 2 ^o 2 



= 0, 







e 4 (o 3 2 



-d B co 4 2 





= 0, 







W 





— e 3 (o 5 2 



= 0, 









d d o) 4 2 



—e^ 1 



= 0. 



Now the equations being homogeneous, only the ratios of the 

 five unknowns have to be found, — that is to say, only four 

 magnitudes are wanted, and for this the first four equations 

 are evidently sufficient. The existence of the remaining equa- 

 tions implies that conditions of consistency have to be fulfilled; 

 and these conditions are at least six in number, for each of 

 the remaining equations determines a ratio already deter- 

 mined, and does so in terms of coefficients not previously met 

 with. Thus equation (6) determines o 2 2 /(o± 2 , which has already 

 been got from equations (1) and (2) : hence we have the 

 condition - •> 



