72 REPORT— 1903. 



the existence of which he had based the theorem now known by his name, 

 which he had published in the previous volume of Crelle's Journal.' 

 These equations are the following, in which x, . . . x,„,„ y, . . . y,„„, are the 

 mil values of x, y which satisfy two given equations, y (.r, y)=0, (l>{x, y)=0 



of orders m, n, and R^ is the value of ^^^j — t "; ^ when x=x^, y=yii. 



(Jc-=.\, . . . mn). 



dx . dy dy . dx 



2i-. 



(A) 





^ R, ' ^ R, ' • • • ^ R^ 



He first points out that if m=^n, these equations are ((2ri — 3)) in 



number, linear in -=~-, (^■=1, . . . w^), fi-om which by solving for -^ from 



Ra R;. 



'n?—\ and substituting in the remaining ((2tc— 3))— «^ + 1 = 2((to — 3)) 

 equations we obtain this number of equations of condition among 

 X, . . . .r,„„, y, . . . y„„„ which is the same number as was previously obtained. 

 But it must be noticed that nothing is said in either case about the con- 

 ditions required to ensure the mutual independence of these 2((n — 3)) 

 equations. Their number, in both places, is found by a ' method of 

 counting constants,' and such a method affords no readily applicable means 

 for dealing with special cases. 



In the next case, in which m-^n, Jacobi again finds the number of 

 equations of condition by counting the constants in equations. The argu- 

 ment is briefly as follows : — An equation of order n (where n<m) is com- 

 pletely determined by {{n)) — \ systems of values of x, y, therefore there 

 must be mn — ((?;)) -j-l equations of condition among mn pairs of quanti- 

 ties if, in accordance with a first hypothesis, mn pairs of values of x, y are 

 to satisfy some particular equation of order n. In accordance with a 

 second hypothesis, these mn pairs of values also satisfy some particular 

 equation of order m. As a consequence of these two hypotheses, more- 

 over, they can always satisfy any equation of order m formed of the sum 

 of this particular one, and of the particular equation of order tc multiplied 

 by any arbitrary factor of order m — n ; and such an equation of order m. 

 will only have ((m)) — ((wi — n)) arbitrary coefiicients (homogeneous) in it, 

 since {{m — n)) can be destroyed by means of the coefficients of the arbitrary 

 factor. Thus there must be an additional number mn — {{m)) + ((m — n)) + 1 

 of equations of condition among the vi7i quantities, since this consequence 

 of the two hypotheses must hold, after the first hypotheses has been satis- 

 fied. By addition, therefore, the total number of equations of condition 

 among the '2mn quantities which satisfy two equations of orders m and n 



' Tlteoremata nova ajgelraica circa systema duarnm cequationnm inter duas 

 variabiles projjositarum . Crelle, vol. xiv. pp. 281-288 ; Worlis, vol. iii. pp. 285-294. 



