118 General Equation of Differences of the Second Order. 



It is readily made manifest, however, that the last determi- 

 nant is expressible in terms of c 4 , c 3 , . . . and continuants which 

 are minors of the coefficient of u x . We thus obtain finally 



K / h b 3 h h ) +bK ( h h h \ 



\a 5 a 4 a 3 a 2 a x J 1 \a 5 a± a 3 a 2 J 



+ c K( h * *> b > ) +6l K( h h ) + ... + * 



\a 6 a 4 a 3 a 2 J \a 6 % a B J 



6. As an illustration of the serviceableness of this new result, 

 let us take the equation of differences, 



to which we are led in trying to find the number (u x ) of 

 arrangements of a set of x things, subject to the conditions 

 that the first is not to be in the last or first place, the second 

 not in the first or second place, the third not in the second or 

 third place, and so on, — a problem which turns up in Professor 

 Tait's well-known investigations in the Theory of Knots 

 (Trans. Roy. Soc. of Edinb. xxviii. p. 159). 

 Putting, in the first place, 



u a} zzxv x ^ 1 + (-l) x ^\x-2) ) .... (a) 



we find, after the requisite reductions, the simpler equation of 

 the same kind, 



Vi = («-lK-2 + V3 + ("-in^-3). 

 The general result applied to this gives 



^-i = K(3, 4, . . . , #— l)v 2 + K(4, 5, . . . , #— 1>! 

 + l.K(4,5,...,^-l)-2K(5,6,...,^-l) + ... + (-l)>-3 



But it is at once evident from the statement of the problem 

 that w 2 = and a B —l; so that, bearing (a) in mind, we must 

 put v 2 = and ^ = 0. The desired solution thus is 



^=^K(4, 5, . . . , #— 1) — 2#K(5, 6, . . . , x 1) 



+ 3xK(Q,7,...,x-l)-... + (-iy(x 2 -4x + 2). 

 For example, 



w 6 = 6K(4,5)-12K(5) + 14=(126~60) + 14, 



= 80, 



as it should be (see Proc. Roy. Soc. of Edinb. ix. pp. 382- 

 391; xi. pp. 187-190). 



7. Clearly the method is applicable to equations of the third 

 and higher orders ; the resulting determinants, however, must 

 then be more numerous and less simple in form. 



Beechcroft, Bishopton, N.B., 

 December 29, 1883. 



