of Edinburgh , Session 1885 - 86 . 
575 
ensuite, au lieu de + 1 a 2 b?cMJ > e i ecrivez + ( 1 a. 2 b. 3 c).{ 4: d. 5 e) ; 
au lieu de - l a 2 b^d. i c. b e i ecrivez - ( 1 «. 2 6. 4 c).( 3 c?. 5 e'), et ainsi 
de suite ; et en egalant k zero la somme de tous ces termes, 
vous formerez l’equation de condition demandde. 
“Je suppose que vous ayez six Equations, combinez les 
termes + abode - &c., relatifs & cinq equations avec la lettre f, 
en observant 1° de n’admettre que les termes oil e precede /; 
2° de changer de signe lorsque / change de place : donnez 
ensuite 1 pour indice & la premiere lettre, 
“ Si vous avez sept Equations, combinez les termes + abode/ 
- &c. relatifs k six equations avec la lettre g de toutes les 
manieres possibles; pour huit Equations, combinez les termes 
relatifs & sept avec la lettre h, en n’admettant que les termes 
dans lesquels g precede h, et ainsi du reste.” 
The really important point in all this is in regard to the manner 
in which the letters are brought into combination. It will be seen 
that the set begun with is abc , consequently a precedes b, and b 
precedes c throughout : then d is combined in every possible way 
with this : e is combined subject to the condition that d precede e ; 
f is combined subject to the condition that e precede /: g is com- 
bined in every way possible : h is combined subject to the condition 
that g precede h : and so on. It would appear therefore that the 
letters w r hich are to be combined in every possible way are d and 
every third one afterwards , and that each of the other letters is con- 
ditioned to be preceded by the letter which immediately precedes it 
in the original arrangement abcdefghi. . . . Condensing these 
directions after the manner of the former case, we should draft the 
rule as follows : — 
(a) Find every possible arrangement of abcdefghi . . . subject to 
the conditions that in each arrangement we must have a, b, c in their 
natural order ; d, e, f in their natural order ; g, h, i in their natural 
order ; and so on. 
(b) Prefix to each arrangement its proper sign in accordance with 
Cramer’s rule. 
(c) Append in order the indices 1, 2, 3 .... to the letters of 
each arrangement. 
(d) Change a m b n c r into (a m .b n .c r ) , d a ej y into {d s .e x .f y ) , &c. 
