400 Mr. J. J. Sylvester on a remarkable Discovery in the 



Hence 



di/. 

 Again, 



^ - (f'i +^'iJ«^(^' 2^) +^(V^-ry')*. 



d ^^ d d 



Hence 



(4)'"-;5-''-''"'"(;r)'^+<'""+"-"''") 



&c. = &c. 



©■•""•-•Gt) ■''•^--'■'(|r)'-.|r.P+ <^~ 



But P' being of c dimensions in o^' and y', and also in a? and y, 

 each of the equations above written will be of l dimensions in on 

 and y, and of no dimensions in a', y' ; in fact, the successive 

 terms of the right-hand members of the above t + 1 equations 

 will be multiples of the (i + 1) quantities 



Consequently a linear resultant may be taken of 



\d^)'^' W ^ ^t; ^^•" V^t;/-^^ 



treating a^^, a^'" ^ .y', . . . y'* as independent, and as the quantities 

 to be eliminated ; and this, according to a well-known principle 

 of elimination, will prove the linear resultant o^ the forj^^oing 

 equations to be equal to the linear resultant of , ' . ':' ' , 



■~ (J^-^.&-'&-(£)-^. ' 



