132 Prof. Sylvester on Elimination mid Derivation 



oxide of iron. If therefore one of these sulphates is mixed with 

 the latter in solution, by far the greater portion of the iron 

 may be precipitated by saturating, as nearly as possible, the 

 solution with potash, and the remaining portion of iron may 

 be thrown down by dilution and boiling. It remains only 

 to be observed that no other acid than sulphuric acid must 

 be present, and that the solution nearly saturated" must be 

 diluted with at least twice or three times its quantity of 

 water. 



This method may not only be employed with advantage in 

 preparing pure oxide of cobalt, but also in analysis. When 

 no error has occurred, the iron is perfectly free from cobalt, 

 although the cobalt may sometimes contain a slight trace of 

 iron. 



During the preparation of pure oxide of cobalt from the 

 roasted ores, arsenious or arsenic acid is constantly present. 

 This need not first be separated by sulphuretted hydrogen, 

 for it is precipitated, on treating it in the manner above de- 

 scribed, as arseniate or arsenite of iron. It is, however, better 

 in this case to add to the solution previously to saturation a 

 quantity of the sulphate of the peroxide of iron, as other- 

 wise there might not be a sufficient quantity of iron present 

 to take up the whole of the arsenious acid, and then arseniate 

 or arsenite of cobalt would also be thrown down. 



XXIII. A Method of determining by mere Inspection the de- 

 rivatives from txw Equations of any degree. By J. J. Syl- 

 vester, i^ii^.S. and R.A.S., P)-qfessor of Natni-al Philosophy 

 in University College^ Lo)idon.* 



T ET there be two equations, one of the nth, the other of the 

 -" W2th degree in x ; let the coefficients of the first equation 

 bean an-\ Un-^i ^0 5 ^^^cA power of .r having a co- 

 efficient attached to it, «„ belonging to x^ and Oq to the con- 

 stant term. 



In like manner let 



bm hm~\ ^0 be the coefficients of the second 



equation. 



I begin with 



A Ride for absolutely eliminating [x). 



Form out of the {a) progression of coefficients [m) lines, 

 and in like manner out of the (Z*) progression of coefficients 

 form [n) lines in the following manner : 



* Conanmnicated by the Author. See the December and January Num- 

 bers of this Magazine. 



