42 Prof. Sylvester 07i Derivation of Coexistence. 



and similarly 



^FBioab ... k) X + ^PD {obc ... I) t - 0. 



r^VD [obc ...I) 

 ^PD(aoc ... I) 

 ^Vjy{abo... I) 



y 



Hence 



)- are severally as -{ 



tJ ^^VT>{abc...o) 



This is the symbolical representation as a formula of the 

 remarkable me/hod 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, x, 7/, z, 



^VD{oacl...l)x+^FD{obd...l) Tz-i-^VB {o cd...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.) Subcorollary to Corollary {S .) 



If there be any number of bases {aba ... /), and any other 

 two fewer in number [fg ... k) 



^ PD (afg ... k) X ^ PD (6 c ... /) 

 + ^VD{bfg.../c) x^FDiac.l) 

 + ^FD{afg...k) x^FB{bc...l) 



The cross is used 

 to denote ordinary 

 algebraical multipli- 

 cation. 



+ i;FB[lfg...k) x^VB{abc...) =0,J 



a formula that from its very nature suggests and ^rou^5 a wide 



extension of itself. 



In conclusion I feel myself bound to state that the principal 

 substance o^ co7~ollaries (1), (2) and (3) maybe found in Gar- 

 nier's Anal;yse Algebrique, in the chapter headed "Deve- 

 loppement de 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. 



FMd 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 



