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



must substantially continue to exist, becomes, and in an accele- 

 rated degree, less and less apparent. 



Thus, take the simple case of a cubic function of 3 variables, 

 and let us confine ourselves to the consideration of the conditions 

 which must be satisfied when it loses a single order. Let U be 

 written out at length, 



aa^-^bf-\-cz^ + Shyz^ + Sizx^ + 3>y« + Sh^z + Si'z^x 

 + Sj'x^y + Qmxyz. 

 The matrix formed out of the coefficients of the linear derivatives 

 becomes 



« .;' 



m 



b 

 h 

 h' 

 m 



J J 



t 



h' 

 c 

 h 



m 



Now by the homoloidal law, if the terms in this rectangle were 

 all unlike, the number of full determinants (3 terms by 3 terms) 

 whose evanescence (except for special values) determines the 

 evanescence of all the rest, should be (6 — 3 + 1) (3—3 4-1), i- e. 

 4 ; but in the actual case, since the evanescence of all the full 

 determinants is a necessary consequence of the function becoming 

 a cubic function of 2 orders {i. e. breaking up into the product 

 of 3 linear functions of x, y, z), and as this decomposability, as 

 is well known, implies only the existence of 3 affirmative condi- 

 tions, the four full determinants 



I 



h<; 



c 



J 



m 



h 



f 

 b 



m 



A'; 



a f 

 f J 



h'; 



m- 



which in the general case would be entirely independent, in this 

 case cease to be so ; and the vanishing of 3 of them must draw 

 along with it by necessary implication (except for special values) 



* That is to say, a syzygetic relation must connect these four determi- 

 nants. I may as well here repeat, that when the vanishing of a set of («) 

 rational integral functions necessarily, and without cases of exception, im- 

 plies the vanishing of another rational integral function, then this function 

 is termed a syzygetic function of the others ; and some power of it must be 

 expressible under the form of a sum of (i) binary products of rational integral 

 functions, one factor of each of which products must be one of the (i) given 

 functions. When the vanishing of all but one of a set of functions in 

 general necessarily implies the vanishing of that one, but subject to cases 

 of exception for specific values of the variables, then it can only be affirmed 

 that the functions of the set are in sj'zygy ; that is to say, that the sum of 

 the products of each of them respectively by some rational integral function 



