530 Mr. S. Roberts on the Order of the Conditions that 



has been made the negative electrode, the rate of rotation is seen to 

 diminish as before in the jar with the brass rod, — a result which 

 is due in this case also to the magnetization condensing the jet, 

 and so increasing the resistance, whereby the intensity of the 

 electric discharge is lessened. 



I am fully aware that further study of these phenomena is 

 still required, and I hope not to be long before I set about it. 

 What I now publish is only a first attempt ; and my principal 

 object in making it known is to draw the attention of physicists 

 to this interesting subject. 



LXV. Note on the Order of the Conditions that an Algebraical 

 Equation may have a set of Multiple Roots. By Samuel 

 Roberts, Esq.* 



"V\7E may find the order of the conditions, in the coefficients, ' 

 X X that an equation may have a set of t multiple roots of 

 the orders n\, ?? 2 , . . . n t respectively in the following manner. 



Let <f> m (x) =0 be the given equation of the degree m; then, 

 in order that it may have m — 2ft given roots denoted by x [} w<# . ., 

 we must put 



(f> m {x x ) = 0, $ m (a? ft ) = 0, $ m (x m _ %n ) = 0, 



and we can make the remaining 2ft roots what we please. 



Now we may take any t of the roots x l3 x q , . . . and may as- 

 sume 2ft quantities such that n x of these are approximate («'. e. 

 indefinitely near) to. one of the t selected roots, ra 2 , in like man- 

 ner, approximate to another of the selected roots, n 3 to another 

 of them, and so forth, until the t roots are exhausted. In this 

 manner we obviously get an equation having m— -2ft given roots, 

 and having a set of t roots of the orders of multiplicity n v n%, 

 n 3) ... n h by giving to the remaining 2ft roots the values so as- 

 sumed. 



But it is plain that these 2ft roots may be assumed approxi- 

 mate to the given roots in a variety of ways, the number of 

 which, however, is perfectly definite. 



For instance, let us consider the n 1 roots approximate to a 

 given root. Although approximate, these n x roots may be con- 

 sidered as greater or less than the given root by an infinitely 

 small difference. Thus if the given root is x v then a^+w, 

 x l — 2v, where w is indefinitely small, give two different manners 

 of approximating to x v We may now distinguish the ways in 

 which we can take ft, roots approximate to x } as follows : — Let 

 die mean a root which approaches x } with a decrement, i k a root 



* Communicated by the Author, 



