of Multiplicity in an Algebraic Equation. 137 



It is easy to see that 



^1 — ^2' pai'tial multiplicities, are less than 2, i. e. are each 



units. 

 h — ^s^ partial multiplicities, will be less than 3, and 

 therefore either I or 2 in value respectively, and so on till we 

 come to 



ly_^ _ /,'^ which will severally be between zero and r— 1, 

 and k^ — of values intermediate between zero and r. 

 Hence there will be 

 /c^ — 2h + kQ multiplicities each of the value 1. 



^2 — 2^'3 + /i'4 ... ... ••• 2. 



fcr-i — 2 kr ... of the value r—1. 



and /t^ of the value r. 



. , d, fx . , , 1 dfx . , d^fx 



In place of/x with -j^ we might employ -J^ with — - 



and so on for; the rest; the values of k^ k.^ ... k^ will remain 

 unaffected by this change; but the former method would be 

 more expeditious in practice. 



The total multiplicity is, of course, = ky. 



Suppose now that we propose to ourselves the converse 

 problem to determine the conditions that an algebraic equa- 

 tion may have a given amount of multiplicity distributed in a 

 given manner. 



If //i 7^2 /'3 ••• K Ije used to denote the given number of 

 partial multiplicities which are respectively of the values 1 2 

 3 ... r, it is easy to see that the quantities derived above by 

 Ici ^2 ••• ^'^r are respectively equal to 



7i, + 2/i2 + + rhr 



ky + 2hl + + rh^^i 



hi + 2//4 + + rhr-i 



hr'. 



dfx 

 Now from ~ — having a factor of the degree h\ common 



dfx 

 with y.r we obtain A:i conditions from — ^ having a factor of 



the degree /tj common with_/, x we obtain kc, more, and so on. 

 So that altogether we obtain in this way 



ky + /c.^ + kr conditions. 



But it may easily be seen that the total multiplicity being 

 k, the number of conditions need never to exceed Z-, in number, 

 no matter what its distribution may be. Hence, besides the 

 enormous labour of the process, and the extreme complexity of 



