42 
Prof. Sylvester on Derivation of Coexistence, 
and similarly 
{o ah 
X 
y 
Hence 
A) ^ f PD [phc ... Z) ^ =s 0. 
f PD [oh c ... V) 
^VD[aoc ... T) 
f PD [ah o ... 1) 
> are severally as <J 
t _> 
L f PD [ah c ... o) 
This is the symbolical representation as a formula of the 
remarkable method discovered by Cramer, perfected by Be- 
zout and demonstrated by Laplace for the solution of simul- 
taneous simple equations. 
Art. (13.) Cor. (4.) In like manner if the number of re- 
peated terms be two greater than the number of equations, we 
have for the relation between any three of them, taken at 
pleasure, for instance, y, 
fPD (o«cZ...Z)jr+f PD [ohd.,,1) y+?PD [o c d...l) z =: 0, 
And in like manner we may proceed, however much in ex- 
cess the number of repeated terms (unknown quantities) is 
over the number of equations. 
Art. (14.) Suhcorollary to Corollary (3. 
If there be any number of bases [ah c 
two fewer in number [fg ... k) 
f PD [a f g ••*k) X f PD [he ,,, 1) 
4- ^ PD [hfg ... k) X f PD (a c ... 1) 
+ ^VD[afg..,k) X ?PD [hc..,l) 
Z), and any other 
The cross is used 
to denote ordinary 
algebraical multipli- 
cation. 
+ X ?PD(«Sc...) =0,J 
a formula that from its very nature suggests and a wide 
extension of itself. 
In conclusion I feel myself bound to state that the principal 
substance of corollaries (1), (2) and (3) maybe found in Gar- 
nier’s Analyse Algehrique^ in the chapter headed ^‘Deve- 
loppement cle la Theorie donnee par M. Laplace, &c.” But 
I am not aware of having been anticipated either in the fertile 
notation which serves to express them nor in the general the- 
orems to which it has given birth. 
End of Part (2). 
[The subject to be continued.] 
University College, London, Dec. 9, 1839. 
P.S. I shall content myself for the present with barely 
enunciating a theorem, one of a class destined it seems to the 
