Mr. J. E. Drinkwater on Simple Elimination. 25 



(2.) Let the equations be 



A, + Xi^' + Yij/ + Z,z + T,^ + ... (m) = 

 Aa + Xa^ + Y^y + Z^z + T^t + ... (ji) = 

 :A„ + X„x + Y„i/ + Z„z + Tj + ... («) = 

 the number of equations and of unknown quantities being 7i, 

 and X;„ representing the coefficient of x in the mth equation. 



(3.) Write down the series of natural numbers 1 2 3 4 ... 7i, 

 and underneath it all the permutations of these n numbers, 

 prefixing to each a positive or a negative sign according to the 

 following condition : 



Any permutation may be derived from the first by consi- 

 dering a requisite immber of figures to move from left to right 

 by a certain number of single steps or descents of a single 

 place. If the whole number of such single steps necessary to 

 derive any permutation from the first be even, that permuta- 

 tion has a positive sign prefixed to it ; the others are negative. 

 For instance, 42 13 ... n may be derived from 1234- ... n, by 

 first causing the 3 to descend below the 4, requiring one single 

 step ; then the 2 below the new place of the 4, another single 

 step ; lastly, the 1 below the new place of the 2, requiring two 

 more steps, making in all four. Therefore this permutation 

 requires the positive sign. 



(4.) The same permutation may be derived in various ways, 

 and it is necessary therefore to show that this rule is not in- 

 consistent with itself: thus the same permutation 4213 ... n 

 might have been obtained by first marching 1 through three 

 places, then 2 through two; and, lastly, 3 through one, making 

 six in all, an even number as before. Without accumulating 

 instances, it is plain, if q be the smallest number of steps by 

 which any number p reaches the place it is intended finally 

 to occupy in that permutation, that if p should advance in 

 the first instance m places beyond this, it must subsequently 

 return thi-ough 711 places ; or, which is the same thing, it must 

 at a later period of the march, allow m of those which it has 

 passed to repass it, so that it will regain its proper place, after 

 the number of steps has been increased from q toq + 2 7}i, 

 which, by the rule, require the same sign as q. The same 

 reasoning applies to every other figure ; and hence the con- 

 sistency of the rule is evident. 



We may, therefore, derive any permutation from any other 

 by altering the sign of the one chosen, according to the change 

 made in its order, and generally it will be found convenient 

 to derive each from the one immediately jireceding it. 



(5.) We thus get the series +1234 7i 



— 2134 n 



+ 2314 n &c. 



N. S. Vol. 10. No. 55. July 1831. E Take 



