1S8 On the Multiplicity of an Algebraic Equatio7i. 



the results, we obtain by this method more equations by far than 

 are necessary, and it requires some caution to know which to 

 reject. 



In my forthcoming paper (to appear in Phil. Mag. of next 

 month) I shall show, by a most simple means, how without the 

 use of derived or other subsidiary functions, to obtain the 

 simplest equations of condition which correspond to a given 

 distribution of a given amount of multiplicity. 



The total multiplicity, say m, being given in as many ways 

 as that number can be broken into parts, so many different 

 systems of 7n equations can be formed differing each from the 

 other in the dimensions of the terms. 



These systems may be arranged in order so that each in 

 the series shall imply all those that follow it, and be implied 

 in all those that go before, without the converse being satis- 

 fied. 



The subject of the unreciprocal implication of systems of 

 equations is a very curious one, upon which the limits assigned 

 to me prevent me from enlarging at present. It is closely 

 connected with a part of the theory of elimination, which, as 

 far as I am aware, has either been overlooked, or has not met 

 with the attention which it deserves ; I mean the theory of 

 Special Factors. 



An example may make what I mean by these clear. 



Let C be a function (if my reader please) void of x, which 

 equivalent to zero implies two given equations in x having a 

 common root. 



Let C be rid of all irrelevant factors, /. e. let C be the 

 simplest form of the determinant, when the coefficients of the 

 two equations are perfectly independent qualities. Now sup- 

 pose, as is quite possible in a variety o/'to«j/s, that such rela 

 tions are instituted between the coefficients alluded to as make 

 C split up into factors, so that C = LxMxN = 0. 



Only one of the factors, L, M, N will satisfy the condition 

 of the co-existence of the two given equations : the others are 

 clearly, however, not to be confounded with factors of solution, 

 or iri'elevant factors, as they are termed, but are of quite a dif- 

 ferent nature, and enjoy remarkable properties, which point 

 to an enlarged theory of elimination, and constitute what I call 

 special or singular factors. 



I shall feel much obliged to anj' of the readers of your 

 widely circulated Journal, interested in the subject of this 

 paper, who would do me the honour of communicating with 

 me upon it, and especially if they would (between now and 

 the next coming out of the Magazine) inform me whether any 

 thing, and if so how much, different from what is here stated 



