[ 119 ] B.yi 



XX. On the Conditions necessary and sufficient to be satisfied in 

 order that a Function of any number of Variables may be linearly 

 ,. equivalent to a Function of any less number of Variables. By 

 ^J. J. Sylvester, F.R.S."^ 



IN the Cambridge and Dublin Mathematical Journal for No- 

 vember 1850, I defined an order as signifying any linear 

 function of a given set of variables, and spoke of a general func- 

 tion of [n) variables as losing (r) orders when the relation between 

 its coefficients is such that it is capable of being expressed as a 

 function of [n—r) orders only. It will be highly convenient to 

 preserve the same nomenclature for the purposes of the present 

 investigation. 



Dr. Otto Hesse, in a long memoir in Crelle's Journal, the 

 contents of which have been described to met, but which I have 

 not yet been able to procure, has given a rule for determining 

 the analytical conditions for the loss of one order. I propose 

 to give a more simple and comprehensive scheme of conditions 

 than Professor Hesse appears to have discovered, applicable not 

 to this case only, but to that of the loss of any number whatever 

 of orders, and shall moreover show in what relation the substi- 

 tuted orders stand to the given variables. 



Dr. Hesse^s rule had been previously stated by me in the 4th 

 section of my Calculus of Forms (Camb. and Dub. Math. Journ. 

 May 1852 J) as applicable to the case of a general function of 

 the 3rd degree of 3 variables becoming the representative of 3 

 right lines diverging from the same point, which is the case of a 

 cubic function of 3 variables becoming a function of 2 linear 

 functions of these variables, that is to say, losing one order : 

 this, perhaps, might have been noticed in the Professor's 

 memoir. I gave also another rule for the same case ; but the 

 true fundamental scheme of conditions about to be set forth will 

 be seen to embrace as mere corollaries all such and such like 

 rules, which in fact supply more or less arbitrary combinations of 



* Communicated by the Author. 



t A distinguished mathematical friend in Paris communicated to me 

 with great admiration Professor Hesse's result over night. I ventured to 

 affirm that, to one conversant with the calculus of forms, the problem 

 could offer no manner of difficulty. An hour's quiet reflection in bed the 

 following morning, or morning after, sufficed to disclose to me the true 

 principle of the solution. 



X Vide vol. vii. p. 187- " When U represents a pencil of 3 rays meeting 



in a point, ^ =0, ^ =0, &c., and also therefore T=0 " (S and T being 



the 2 Aronholdian invariants of U, and a, b, c, &c. the coefficients of U) ; 

 " also in place of this system may be substituted the system obtained by 

 taking all the coefficients of the Hessian zero.'* 



