120 Mr. J. J. Sylvester on Polynomial Functions which 



the conditions, rather than the naked conditions themselves in 

 their simple form and absolute totality. 



I shall call the function to be dealt with U, and shall consider 

 U to be a homogeneous'^ rational function of m dimensions in 

 respect of or,, a-j, . . . Xn, and shall inquire what are the conditions 

 which must obtain when U is capable of being expressed as a 

 function of only (n—r) orders, say /j, /g, . . . 4 -r^ each of which 

 is of course a homogeneous linear function of the given (n) 

 variables. 



Let the term derivative of U be understood to mean any result 

 obtained by differentiating U any number of times with respect 

 to one or more of the variables x^^ x^ . , ,Xn. The first deriva- 

 tives will be of (m — 1) dimensions, the second derivatives of 

 {m—2) dimensions, and so on; and finally, the (m — l)th deriva- 

 tives will be homogeneous linear functions of x^, x^, . . .Xn* Sup- 

 pose U to be expressible as a function of l^, 4> • • • ^n-r- It is 

 immediately obvious that the derivatives from the 1st to the 

 (m~l)th inclusive will be all expressible as homogeneous func- 

 tions of /„ /gj • • • ^n-vf and vanish when these vanish. But this 

 statement is in substance pleonastic ; for by means of Euler's 

 well-known law, any derivative of U, say K, may be expressed 

 (to a numerical factor pres) under the form of 

 dK dK dK 



^'dx.'^^^'dx^'^ ••• "^"""'^^ 



and consequently, whenever the linear derivatives of U vanish, 

 all the upper derivatives of U, including U itself, must vanish at 

 the same time. The number of these linear derivatives, say v, 

 will be the number of terms in a homogeneous function of (n) 

 variables of {m — l) dimensions, that is to say, 



n . (n—1) . . . (n—m + 2) 

 1.2 . .. (m-1) • 



Again, if all the v linear derivatives vanish when the (n — r) 

 equations 1^ = 0, 1^=0, .. . l^-r^O are satisfied, r being greater 

 than zero, this can only happen by virtue of these v derivatives 

 being linear functions of {n—r) of them. Now, conversely, 1 

 shall prove, that if it be true that all the linear derivatives of U 

 are linear functions (n — r) of them, then U may be expressed 

 as a function of these {n—r) ; and this rule, as will be immedi- 

 ately made apparent, will give the necessary and sufficient con- 

 ditions for the loss of r orders in the most simple and complete 



* It is a common error to regard homogeneity of expression as merely 

 a means for satisfying the desire for symmetry ; the ground of its apph- 

 cation and utiUty in analysis lies, in fact, much deeper ; it is essentially a 

 method and a power. 



