the Dialytic Method of Elimination. 229 



composable into factors, then it is evident that we must have 

 R = R' . R", and consequently that the existence of such a parti- 

 cular resultant as E' will argue the necessity of the existence of 

 another resultant R" ; in other words, the resultant so found 

 cannot be in a strict sense the true and complete resultant for 

 the particular case assumed, and yet the process employed ap- 

 pears to give the complete resultant, or at least it is difficult to 

 see how the wanting factor escapes detection. To make this 

 matter more clear, take a particular and very simple case, where 

 m=2 n=2 p = 2 = 0, so as to form the system of equations 



A'a?2 4-B'^y4-C'/+(D'a? + EV)^=0 I (C.) 

 Ix + my + nz =0j 



By virtue of my theorem, the degree of the resultant R' is 

 2(2 + 2 + 1) —3 . 1 = 7, but the resultant R of the system 



kx^ + Bxy + Qy'^ -A- (D^ + Y^y)z + F^^ ^ q^ 

 Mx^ + B'^y + Cy 4- (D'.2? + EV)^ + r V^ = I (D.) 

 lx-\-my-\-n2 =0j 



which becomes identical with the former when r=0, F' = Ois of 

 2x2 + 2x1+2x1, 2. e. of 8 dimensions. Hence it is evident 

 that whenF=0, F = 0, R must become R'xR". 



It will be found in fact, that on the supposition of F = 0, F' = 0, 

 R becomes equal to w x R" ; and accordingly, besides the portion 

 R' of the resultant of system (C), found by the method in the 

 text, there is anotRer portion n which has dropped through ; but 

 it may be asked, is n truly a relevant factor ? were it not so, the 

 theory of the resultant would be completely invalidated ; but in 

 truth it is; for n = will make the equations in system (C.), con- 

 sidered as a particular case of system (D.), capable of coexisting; the 

 peculiarity, which at first sight prevents this from being obvious, 



consisting in the fact that the values of -, -^ which satisfy the 



Zi z 



three equations when /i = become infinite. 



Thus, finally, we have arrived at a clear and complete view of 

 the relation of the particular to the general resultant. 



The general resultant may be zero, in which case the particular 

 resultant is something altogether diff"erent from an ordinary re- 

 sultant j or the particular resultant may be a root of the general 

 resultant, or it may be more generally the product of powers of 

 the simple factors, which enter into the composition of the general 

 resultant ; or lastly, it may be an incomplete resultant, the fac- 

 tors wanting to make it complete being such as when equated to 

 zero, will enable the components of the resultant to coexist, but 



Fhil Mag. S. 4. Vol. 2. No. 10. ^ept. 1851. R 



