Prof. Sylvester's Theory of certain Multiple Roots. 249 



oxygen of which combines with the zinc when the pole is of 

 that metal, or with hydrogen of the acid of the salt when 

 platinum is used. With chlorides of potassium and calcium, 

 there was also a deposit of oxide of zinc at the positive pole, 

 and sometimes a part of the oxide of zinc was taken up by 

 the acid of the hydracid salt and carried to the negative side. 



I conceive that a sufficient number of cases has been in- 

 vestigated to warrant the general conclusion, " that when 

 aqueous solutions of primary combinations of elementary 

 bodies are submitted to voltaic agency, the dissolved substance 

 is not decomposed by the current, but only the solvent." 

 The rule of course does not embrace combinations of the se- 

 cond order, such as oxysalts, which as every one knows 

 are resolved into their constituent acid and alkali under vol- 

 taic agency. 



[To be continued.] 



XLII. A nciv and 7iiore general Theory of Multiple Roots. 

 By J. J. Sylvester, F.R.S. S^-c., Professor of Natural Phi- 

 losophy in University College, London^. 



T SHALL begin with developing the theory of polynomials 

 containing perfect square factors, one or more. 



First, let us proceed to determine the relations which must 

 exist between the coefficients of such polynomials, and after- 

 wards show how they may be broken up into others of an in- 

 ferior degree. 



A parallelogram filled with letters standing in one row is 

 intended to express the product of the squared difference 



of the quantities contained. Thus Q a b^ indicates (« — J)^ 

 (abc^ is used to indicate (a — b^) . {a—c)~ . {b — cY, and so 



forth. 



Suppose now that two of the roots e^ e^ ... e„ belonging to 

 the = "f-v = are equal to one another, it is clear that 



( Ci Co ... ^n ^ = ; and moreover is a symmetric function, 

 and can be calculated in terms of the coefficients o^ f x. 



Next let us suppose that we have two couples of equals, (as 

 for instance a and b, two of the roots equal, as also c and d 

 two others) it is clear, that on leaving any one of the roots 



out the {n 1) ihat are left will still contain one equality, and 



therefore we have 



CJ^'ej::: e„ ) = O ( ^i^a "• gn ) = '"( e, e^ ...'g~ ) = 0. 

 ♦ Communicutcd by the Author. 



