the Dialytic Method of Elimination. 227 



indicated upon this supposition. If we take m=:n=p = qj and 

 t = t' = &c. ={i)=m — e, the exponent of the total dimensions of 

 the resultant becomes 



(r — l)rw— r(r — 2)(m — e) 

 =srm—r{r—2)ej 



when 6=0, this becomes mr, which is made up of 2m units of 

 dimension belonging to the coefficients of the first column, and 

 of m belonging to each of the (r — 2) remaining columns. Con- 

 sequently, if we have 



Gm (cc, i/)-\-rjX-\- r)'X = 

 Hm(.r,y) + ?X + 5^X' = 



or any other number of equations similarly formed, the result of 

 the elimination is always of m dimensions only in respect of 

 ^) V} ?> ^j» or of f ', 77', f , 6', and of 2m in respect of the coeffi- 

 cients in F, G, H, K. 



I now proceed to state and to explain some seeming paradoxes 

 connected with the degree of the resultant of such systems of 

 defective functions as have been previously treated of in this 

 memoir, as compared with the degree of the general resultant of 

 a corresponding system of complete functions of the same number 

 of variables. 



In order to fix our ideas, let us take a system of only three 

 equations of the form 



Fm(.r,7/) + F^_,(^,y)/=0^ 



Gn[a;,y) + Gn-c{a:,y)/ = 0^. . • • (B.) 



Up {oc,y) + llp-i (^,?/)/==0^ 

 The resultant of this system found by the preceding method is 

 in all of 2m-\-2n + 2p — ^(, dimensions. But in general, the 

 resultant of three equations of the degrees m, w, p is of mn -{- mp 

 -\-np dimensions. 



Now in order to reason firmly and validly upon the doctrine 

 of elimination, nothing is so necessary as to have a clear and 

 precise notion (never to be let go from the mind^s grasp) of 

 the proposition that every system of [n) homogeneous func- 

 tions of (n) variables has a single and invariable Resultant. 

 The meaning of this proposition is, that a function of the co- 

 efficients of the given functions can be found, such that, when- 

 ever it becomes zero, and never except when it becomes zero, 

 the given functions may be simultaneously made zero for some 

 certain system of ratios between the variables. The function so 



