122 Mr. J. J. Sylvester's Enumeration of the 



be a syzygetic* function of all the rth. minors of any other form 

 of the same ; and consequently a function of \ of any degree^ 

 whether all its factors be or be not distinct, which is common to 

 the rth minors of one form of U + \Y, will remain so to the rt.h 

 minors of any other form of the same. 



The law exhibiting the connexion of each ?'th minor of one 

 form of W (any homogeneous quadi-atic function) with all the 

 ?-th minors of any other form of W, will form the subject of a 

 distinct communication. 



Finally, to fully comprehend the annexed discussion, the fol- 

 lowing principle must be apprehended. 



If any factor K" enter into all the rth minors of W, and if K' 

 be the highest power of K common to all the (r+l)th minors, 

 then K^'"* will be a common factor to all the (r— l)th minors. 



Let r be taken unity ; it is easily provedf that the complete 

 determinant of any square matrix may be expressed by the dif- 

 ference between two products J, each of two first minor determi- 

 nants decided by a certain second minor determinant. The pro- 

 portion is therefore demonstrated for this case, and thereby in 

 fact implicitly for eveiy case, inasmuch as the first minors of 



* If A=j[)L+5rM + rN + &c., where p, q, r . . do not any of them be- 

 come infinite when L, M, N . . or any of them become zero, A may be termed 

 a sj'zygetic fimction of L, M, N. . . 



in the theorem above alhided to, it will be shown (as might be expected) 

 that the syzygy in the case concerned is of the simplest kind, i. e. that each 

 rth minor of a quadratic fimction of any number of letters is a homogeneous 

 linear function of all the rth minors of the same quadratic ftmction linearly 

 transformed. 



f This will appear in my promised paper on Determinants and Quadratic 

 Fimctions. 



J When the matrix is symmetrical about one of its diagonals (as it is in 

 the case which we are concerned with), one of these j)roduct8 becomes a 

 square. I may take this occasion of hinting, that the theory of quadratic 

 functions merges in a larger theoi-y of binary fimctions, consisting of the 

 sum of the midtiples of binary products formed by combining each of one 

 set of quantities, x, y, z . . . with each of the same number of quantities of 

 another set, as x', y', z'. . . For instance, 



axx'+bxy'-\-cxz' 

 -^a'yx'+b'yy'-^-c'yz' 

 + a"zx'+b"zy"+c"zz" 



would be a binaiy function, and its determinant (no longer, as in a quadratic 

 function, s^Tumetrical about either diagonal) would correspond to the square 

 matrix 



a b c 



a' b' c' 



a" b" c". 



Almost all the properties of quadratic apply, with slight modifications, to 

 binary functions. 



