A Special Determinant. 



373 



minants so differently arrived at in (VII.) : and that, generally, the 

 co-factor of < in i N,.D,.+i | is 



e, 



«! 





. 



... 



e. 



a. 



a. 



. 





0. 



a. 



a._ 



a. 



... 



e,_, 



a,_. 



a,_. 



a,_3 ... 



... a. 



e. 



a,. 



a,_. 



a,_._ . . . 



... a^ 



(XII) 



4. In the special case where the original fractions are not only 

 equal, but equal to zero, and where therefore a^a, = bl, the G's may 

 be altered into 



a, { {a„ a,'^a„ a,) - {b„ b;^b„ b,) \ , 



cti { (^o> <^n ^^'^zfe* <^2. <^i) - (b„ &2, b^'$b^, b^, b,) } , 



and consequently the requisite conditions take the form of the 

 vanishing of n (not 7i - 1) expressions of one and the same form, 

 namely, 



{a:$ia,)-{b:$b,), 

 {a^, a;§a,, a^)-{b„ b;^b„ b,) (xiii) 



If we further specialise by putting <Xo = 2, the n equations to be 

 satisfied may be written 



(1, aJa„l)-(MA) = 0, 

 (1, «!, a.\a._, a„ 1) - {b„ b^b^_, b,) = 0, 



(XIV) 



One solution of those last equations is got if a„ a„ ... be made 

 equal to the even-numbered members, and b„ &,, ... to the odd- 

 numbered members of the series 



111 1 



1!' 2!' 3!' ••" {2n) V 

 as is seen from the vanishing of 

 1 



(2m) ! 



1 — ^2m, I . ^ + ^2m, 2 ' ^^ 



' (2m) P ^ 



