228 Mr. J. J. Sylvester on Extensioiis of 



found, which is suflBcient and necessary to condition the possi- 

 bility of the coexistence of the equality to zero of each of the 

 given functions, is their resultant, and by analogy they may be 

 termed its components. It follows that if R be a resultant of a 

 given system of functions, any numerical multiple of any power 

 of R or of any root of R when (upon certain relations being sup- 

 posed to be instituted between the coefficients of its components) 

 R breaks up into equal factors, will also be a resultant. This is 

 just what happens in system (B.) when m — n—p-=L] the re- 

 sultant found by the method in the text is of the degree 3m ; 

 the general resultant of the system of three equations to which 

 it belongs is of the degree 3m* ; the fact being, that the latter 

 resultant becomes a perfect mth power for the particular values 

 of the coefficients which cause its components to take the form 

 of the functions in system (B.). 



Suppose, however, that we have still m = 7i=j9,but l less than 

 (m), 6m — 3t will express the degree of the resultant of system (B.); 

 but this is no longer in general an aliquot part of 3m*, and conse- 

 quently the resultant of system (B.) that we have found is no 

 longer capable in general of being a root of the general resultant. 

 The tmth is, that on this supposition the general resultant is zero ; 



oc u 



as it evidently should be, because the values - =0, - = satisfy 



the equations in system (B.), except for the case of m=(-; conse- 

 quently the resultant furnished in the text, although found by the 

 same process, is something of a different nature from an ordi- 

 nary resultant ; it expresses, not that the system of equations (B.) 

 may be capable of coexisting, but that they may be capable of 



X 11 



coexisting for values of -, - other than and 0. This is what 



I have elsewhere termed a sub-resultant. But there is yet a 

 further case, to which neither of the above considerations will 

 apply. This is when m, n, p are not equal, butjo — 1=0. 



On this supposition the degree of the resultant of B becomes 

 2m 4- 2/1 —p) which in general will not be a factor of mn-^mp + np; 



X 



and in this case it will no longer be true that the values — =0, 



z 



-^=0 will satisfy the system B, inasmuch as the last equation 



therein cannot so be satisfied. Now if we call the general resultant 

 R and the particular resultant R', if R' should break up into fac- 

 tors so as to become equal to (r'j^x (s')*. • • (0^ i^ might be the 

 case that R should equal (/)" . {^f . . . (/')^, and there would be 

 nothing in this fact which would be inconsistent with the theory 

 of the resultant as above set forth ; but suppose that R' is inde- 



