573 
of Edinburgh, Session 1885-86. 
3°. Combining d with each of these terms subject to the con- 
dition that c precede d : result — ■ 
abed — acbd + aedb + cabd - cadb + edab . 
4°. Combining e with each of these terms in every possible way : 
result — 
abode - abced + abecd — aebed + eabed 
- acbde + aebed 
5°. Appending indices : result — 
6°. Changing a m b n into (a m h n - b m a n ), c' r d s into (c r d s - d r c s ). &c. : 
result — 
(a 1 b 2 ~ b 1 a 2 )(c 3 d 4 — c? 3 c 4 )e 5 — (afi 2 — bgx^ipgi^ — d^c 5 )e^ + . . . . 
This is the required resultant in the required form. 
It is of the utmost importance to notice that what is accomplished 
in 1°, 2°, 3°, 4° is simply (a) the finding of the arrangements of 
a, b, c, d, e subject to the conditions that a precede b, and c precede 
d, and obtaining each arrangement with the sign which it ought to 
have in accordance with Cramer's rule. The number of necessary 
directions might thus be reduced to three, viz., (a), (5), (6), in 
which case (1), (2), (3), (4) would take their proper places as 
successive steps of a methodic and expeditious way of accomplish- 
ing (a). 
Laplace appends a demonstration of the accuracy of this de- 
velopment of the resultant of the wth degree, the line taken being 
that if the multiplications were performed the terms found would 
be exactly the 1.2.3 n terms of the resultant, and would bear 
the signs proper to them as such. 
He then goes on to deal with a rule for obtaining a like develop- 
ment in which as many as possible of the factors of the terms are 
resultants of the third degree. 
To do so succinctly he is obliged to introduce a notation for 
resultants. On this point his words are — 
