an Algebraical Equation may have a set of Multiple Roots. 533 



H], ?? 2 , . . . being all different. If p of them Lave a common 

 value, and q of them have another common value, and so on^ 

 we must take for our result 



m . (ro — 1) (m— 2) . . . (w — S/i— t + 1) 

 1.2...;?. 1.2...?... 



Again, if a = and \=1, the order is given by (a) or (b) ; so 

 that in point of fact the coefficient of \2»+f-i in the general ex- 

 pression (c) is given by those formulae. 



Now, in order to obtain the complete expression for the order, 

 in some cases the following consideration suffices : — The expres- 

 sion must remain unchanged when for X, u we substitute 

 \ + wia, —a. It is evident that if instead of beginning with 

 the coefficient of %% and proceeding by increments (or decre- 

 ments, as the case may be) of the orders, we commence with 

 the last term of the given equation, and proceed backwards by 

 decrements (or increments, as the case may be) of the orders, 

 we have \-\-am, — a replacing A, a. 



I take the cases which this consideration enables us to deal 

 with. 



(1) Three equal roots. 

 The order is of the form 



(m-2) {S\* + M\ci + m{m-l)a*\ , 



M remaining to be determined. 



The above expression must be identical with 



(m-2){S(\ + am) q -~M{\ + *m)u + m{m-l)**\; 



.-. 3mV-Mm« 2 = ; 



M = 3m, 



and the required order is 



3 (m — 2)\(X + ma) + m (m - 1) (m — 2) . 



(2) Two equal pairs of roots. 

 The order is of the form 



(rn _ 2) (m - 3) jW + M\« + m ^*l\ , 

 which must be the same as 



(ro-2)(m-8) | 2{\ + *m)*M(X + am)a + m ( m ~ l > \. 

 Phil. Mag. S. 4. No. 226. Suppl Vol. 33. 2 N 



