460 Mr. J. J. Sylvester on a new Theory of Multiplicity. 



Nombres, 3rd edition, vol. ii. p. 438.) But he has overlooked 

 the existence of the equation of condition {£), without which, 

 essentially linked as it is with the irreducibility of the pi^oposed 

 equation, such a system of successive equations could not exist. 

 It appears that Legendre was not himself aware that there was 

 any antagonism between the results at which he had arrived and 

 those of Abel. If, however, the non-existence of the condition 

 (£ ) could without error be assumed, the objection of the learned 

 author of the treatise on the Calculus of Functions in the Ency- 

 clopedia Metropolitana (p. 382) would undoubtedly be applicable 

 to Abel's method. 



Long Stratton, Norfolk, 

 April 14, 1852. 



[To be continued.] 



LXV. Observations on a New Theory of Multiplicity. 

 By J. J. Sylvester, Barrister -at-Law* . 



IN the Postscript to my paper in the last Number of the Ma- 

 gazine, I mis-stated, or to speak more correctly, I under- 

 stated the law of Evection applicable to functions having any 

 given amount of distributive multiplicity. The law may be stated 

 more perfectly, and at the same time more concisely, as follows. 

 Every point represented by the coordinates a v fi v . . <y v for 

 which the multiplicity is m u will give rise in every evectantf of 

 the discriminant of the function to a factor {ct 1 x + l3 l .y + .. 7,*) m r", 

 (n) being supposed to be the degree of the function. Hence if 

 there be r such points, for which the several multiplicities are 

 m v »i 2 , . . rn r , every evectant must contain (»i, + m 2 + • • + m r) - n 

 linear factors; and as the tth evectant is of the degree i.n, it 

 follows that all the evectants below the (m l + m^+ . . +m r )th 

 evectant must vanish completely, and this Evectant itself be con - 



* Communicated by the Author. 



t Frequent use being made in what follows of the word Evectant, I re- 

 peat that the evectant of any expression connected with the coefficients of 

 a given function (supposed to be expressed in the more usual manner with 

 letters for the coefficients affected with the proper binomial or polynomial 

 numerical multipliers) means the result of operating upon such expressions 

 with a symbol formed from the given function by suppressing all the bino- 

 mial or polynomial numerical parts of the coefficients to be suppressed, 

 and writing in place of the literal parts of the coefficients a, b, c, &c. the 



symbols of differentiation — , —, — , &c. ; in all that follows it is the suc- 



da do dc 

 cessive evectants of the discriminant alone which come under consideration. 

 I need hardly repeat, that the discriminant of a function is the result of the 

 process of elimination (clear from extraneous factors) performed between 

 the partial differential quotients of the function in respect to the several 

 variables which it contains, or to speak more accurately, is the characteristic 

 of their coevanescibilitv. 



