572 
Proceedings of the Boyal Society 
cela pose, an lieu de + 1 a. 2 b. 3 c. A d ecrivez 
+ ( 4 a.% - 1 b 2 a).(?c.^d - 3 d. 4: c ) , 
et ainsi des autres termes, et l’equation de condition sera 
0 = Qa 2 b - '%?a).(*cM - 3 dAc) - ( 4 a 3 b - Wa).(Wd - 2 d±c) 
+ ( 4 a. 4 6 - 1 b. 4t d).(*c, 3 d - 2 d 3 c) + ( 2 a 3 b - 2 b 3 a).QcM - 4 d 4 c) 
- ( 2 a. 4 5 - %M).Qc*d - l d 3 c) + ( 3 a. 4 5 - *bM).Q-cM - l d 2 c ) . 
“ Je suppose que vous ayez cinq Equations, ecrivez les six 
termes + abed — acbd + . . . relatifs k quatre Equations, et 
combinez-les avec la lettre e de toutes les manieres possibles, 
en observant de changer de signe chaque fois que e change de 
place; donnez ensuite l’indice 1, &c., &c., ; au lieu 
du terme + 1 a 2 c 3 b. 4: e. b d Ecrivez ( 1 a. 8 5 - 1 b 3 a).( 2 c. b d - 2 d. b c).^e, 
&c 
“ Lorsqu’on aura six equations, on combinera les termes 
+ abcde - abced+ &c., relatifs k cinq equations avec la lettre/, 
en observant 1° de n’admettre que les termes dans lesquels e 
precede /; 2° de changer de signe lorsque /change de place: 
on transformera ensuite, par la rkgle pr4c4dente, . ” 
Notwithstanding the multiplicity of instances, the rule here 
illustrated is not made altogether clear. This is due to two 
causes, — first, the linking of one case to the case before it; and, 
second, the want of explicit notification that the letters b } d, f . . . 
are combined in one way, and the intervening letters c, e, . . . in 
another. For the sake of additional clearness, let us see all the 
steps necessary in the case of the resultant of the five equations 
a r x x + bx 2 + c r x z + d r x± + e r x 5 = 0 (r = 1, 2, 3, 4, 5) , 
and supposing, as we ought to do, that the case of four equations 
has not been already dealt with. These steps are — * 
1°. Combining b with a subject to the condition that a precede 
b ; result — 
ab. 
2°. Combining c with this in every possible way , the sign being 
&c. : result — ■ 
abc - acb + cab , 
