an Algebraical Equation may have a set of Multiple Roots. 531 

 which approaches x 1 with an increment, then we may have 



h 



h 



h • 



• • ? «!> 



d } 



h 



h> 



• • 2 Wl , 



d x 



d 2 



h • 



• • Init 



d, 



d, 



d 3 . 



• • *«x> 



dy 



d 2 



d 3 . 



. . d ni . 



There are therefore n x + 1 ways in which we may constitute #, a 

 multiple root of the order (n^ . For a particular set of / given 

 roots, then, the number of ways in which we can form a set of t 

 multiple roots of the orders n v n^, . . . n t respectively is 



(» l + i)(w 2 +i)..>,+i), 



for which expression we may write Y(n+\). 



Again, the number of ways in which we can take t roots out 

 of m — Xn roots is 



(rn—-'Zri)(m' — Sw— 1) . . . (m—Xn—t + 1), 



counting permutations of the / roots, as in the present case we 

 have to count them. 



Hence the total number of ways in which we can attribute 

 m — Xn given roots and a set of t multiple roots of the orders 

 n x , Bg, ... . n t is 



¥{n + l)(m-2,n){m-Xn-l)...(m-Xn— t + l). ...(«) 



But this is the order which we have to obtain ; for it is the 

 order of the conditions that the given equation <£ m (#)=0 may 

 have the stated set of multiple roots, combined with the condi- 

 tions that this equation may have m—%n given roots. This last 

 set of conditions being linear in the coefficients, the result is 

 simply the order of the conditions for the multiple roots. 



It has been supposed that n v n 2 , . . . n t are all different; if p 

 of them have one common value and q of them have another 

 common value, and so on, we must divide the result by 

 1 . 2 . . .p . 1 . 2 . . . q . . ., so that we have 



P (n + l)(ro--Sft) (m-Sw-1) . . . (m — ^n-t + l) 



1 .2... p. 1.2 . . . q . . . ' ' ( } 



M. Jonquieres has given these results as particular deductions 

 from a theorem of great generality, on the multiple contacts of 



